Angle between vectors - how to calculate and code it!

07 March 2021 - Not one of those ambulance chasers. Her bed, however, was neatly made up and her pajamas were on the hook behind her door. I rushed into the nearest clothing available and hurried downstairs. Jenna sat at the kitchen island, eating a bowl of cold cereal and watching the Weather Channel. This addition is a hundred times worse than the time she changed the bus routes. A thousand times worse than the time she changed the school mascot from a golden retriever to a bulldog. jesse shepard Free Vector cross product calculator - Find vector cross product step-by-step This website uses cookies to ensure you get the best experience. By using this website, you agree to our Cookie Policy.where θ is the angle between /(/vec{a}/) and /(/vec{b}/) Angle Between Two Vectors Examples. Let’s see some samples on the angle between two vectors: Example 1: Compute the angle between two vectors 3i + 4j – k and 2i – j + k. solution: Let /(/vec{a}/) = 3i + 4j – k and /(/vec{b}/) = 2i – j … labster physiology They let me paint for a few hours every day, and my aunt sent endless supplies of pastels and Conte crayons and paper. I learned to walk very quietly around certain inmates, and I made a lot of flattering drawings. That was my first taste of prostitution. I turned it over, looked at the date unbelievingly, and felt a foolish grin take hold of my face. Gerald Andrew Edwards had not been in Scotland on the night our cottage had been ransacked, not unless he had spent the next twelve hours driving very fast. The following morning, he had been nabbed for speeding near Tavistock, about as far from Scotland as England went. I took my gold fountain pen from my bag, made a note of the details, and then, clutching the pen in my hand, followed Alex towards the house. I shrugged off my damp coat and was arranging it over a chair and considering the effort of asking for a measure of coal to make a fire, when I heard a gentle knock. One such had contained a deliberate misspelling that was corrected along the way by some conscientious telegraphist, thus rendering the message totally meaningless.

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Angle between two vectors - OnlineMSchool

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  • The Geometry of the Dot and Cross Products

They said the priest would be happy to ring the bell should there be any need. She stood for a moment observing the group Mooktu and Bister were working with, then spent another moment studying the onlookers-mostly old men and young girls-lining the pavement along the street side of the yard. There were far fewer than the first day, but clearly, people knew their party was still at the auberge. les 7 psaumes de victoire This online calculator finds the angle between two vectors person_outline Timur schedule 2019-06-06 07:47:48 This calculator finds the angle between two vectors given their coordinates.Just like the dot product is proportional to the cosine of the angle, the determinant is proprortional to its sine. So you can compute the angle like this: dot = x1*x2 + y1*y2 # dot product between [x1, y1] and [x2, y2] det = x1*y2 - y1*x2 # determinant angle = atan2(det, dot) # atan2(y, x) or atan2(sin, cos) bcf jackets She froze, her weight steady against him, her thighs spread, her knees on either side of his hips. Breathless, she waited, stunned by the sense of vulnerability that washed over her. A single upward thrust would sink his staff into her, hard against the source of the heat pouring over him from between her widespread thighs. He eased his convulsive grip on her hands and raised them, placing them on the pillow, one above each of his shoulders. Another deep breath allowed him to run his hands back along her arms to curve about her shoulders. Campbell looked around for rooftop snipers, and that seemed almost funny to Peter. A big flamingo pink Hollywood-style house. A marvel how you could smooth out these things in the end-national security matters. Some mail was waiting for her-letters from Damian.

He felt light-headed, a pleasant dreamy torpor pressing him down and down into the infinitely soft mattress. They fell down and others kept on coming. Breathing in rarefied air saps all the strength. doom cheats Online calculator. Angle between two vectors. Definition. The angle between two vectors, deferred by a single point, called the shortest angle at which you have to turn around one of the vectors to the position of co-directional with another vector. Basic relation.This calculator can be used for 2D vectors or 3D vectors. If a user is using this vector calculator for 2D vectors, which are vectors with only two dimensions, then s/he only fills in the i and j fields and leave the third field, k blank. If using this calculator for a 3D vector, then the user enters in all fields. state id lookup The maids who opened the doors to him suggested that a visit in the morning might be more productive, and he reluctantly agreed, although Sunday was often a difficult time to interview persons of this class, for reasons different from those on a week-day. She was well guarded by a butler who looked nearly as old, and a house-maid in her late forties who bore a striking resemblance to the butler. All three had spines as straight and unbending as one of the gleaming brass pokers arrayed beside the ten-foot-wide marble fireplace, and Holmes would never have got inside had he been mere trade. Evidently he thought that the police had brought her. Come down and put an end to this. Meanwhile, I was back next to the officer. I want the police to leave the hotel. Clear the street of those reporter creeps and the patrol cars. At most, I could hold him for illegal possession of a firearm.

Distance between Two 3D Points - Engineering ToolBox

There was another report, and both fell to the ground. Crow, still groggy, had a pistol in his hand and was trying to get up. He swung at her with his left hand, catching her on her cheek. She reversed the weapon and hit him hard on the head. Therefore, angle between two vectors is 6.28 Degrees or 0.11 radians.Taking two vectors, we can write every combination of components in a grid: This completed grid is the outer product, which can be separated into the:. Dot product, the interactions between similar dimensions (x*x, y*y, z*z). Cross product, the interactions between different dimensions (x*y,y*z, z*x, etc.). The dot product ($/vec{a} /cdot /vec{b}$) measures similarity because it only costco online return policy Come, join me in a glass of this marvellous stuff. In this case, however, I think I may have found a buyer for Baskerville Hall. He blurted out, "All that crap about the United States planning to dump poison in the oceans and the Russkies trying to drown us all. But not a word of it was invented, I assure you. Too illogical to argue reasonably and sensibly. Chase shrugged and picked up his fork and carried on eating. The circle of quiet had spread along the trestle tables. Heads were inclined like rows of obedient marionettes.

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Keep in mind that the estate can run itself if it has to. Marsh is trapped there at the moment, but not for reasons that affect you and Gabe. Vector Calculator: add, subtract, find length, angle, dot and cross product of two vectors in 2D or 3D. Detailed expanation is provided for each operation.Calculate the magnitude of three dimensional vectors (3D Vectors) for entered vector coordinates. The 3D vectors are using the x-y-z axes. The vector OP has initial point at the origin O (0, 0, 0) and terminal point at P (2, 4, 5). So we can draw the vector OP . fcpl catalog I could get used to being adored. Mediocre calamari, excellent bruschetta and flawless service so far. Once we were old enough to make our own choices, Connie and I started doing our own holiday thing, just the two of us. Now she has a husband and a munchkin and I get together with them. Pretty much bare-assed, too, if memory serves me right. At least the parts I can remember.

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When I wished to interview a certain Brother Stultus about the early days in England, he was not to be found. One of the brothers told me he had gone to Mexico (we were then in San Diego), but that was some weeks before. Stultus was not a young man, and I was concerned, but there was not much I could do. In a Cartesian coordinate system, the three unit vectors that form the basis of the 3D space are: (1, 0, 0) - describes the x-direction (0, 1, 0) - describes the y-direction (0, 0, 1) - describes the z-direction; Every vector in a 3D space is equal to a sum of unit vectors.Oct 20, 2019 taurus single love horoscope this week Even though its daily gains were now measured in kilometers instead of tens of kilometers, the Second Brigade Tactical Group was still advancing-still drawing South African troops away other fronts. Transshipped by rail the long way round through neutral Botswana, the Third Brigade had shot its way onto South African territory three days after its two counterparts. Confronted by two more immediate threats to its vital northern and eastern Transvaal mining complexes, Pretoria had stripped its border with Botswana of almost every trained man able to bear arms. The sun stood high overhead, beating down mercilessly on grasslands just starting to turn from yellow-brown to a lush, rich green. He did not speak as their gazes locked. Awareness bolted through her like lightning and every inch of her skin tingled as he moved away. It felt as if he was saying it now, again, without words.

I decided to give her until one, and ten minutes before then, she came around the corner, as carelessly dressed as I, but pale, red-eyed, and half-focused. I greeted her with apprehension, wondering what new upheaval had tossed her here in this condition, but she managed a distracted smile and seemed to be trying to pull herself together. Something happened this morning, and I totally forgot the time until Margery reminded me that I had a luncheon date with you. Feb 20, 2013A vector space (also called a linear space) is a set of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called s are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any operations of vector addition and scalar multiplication must jason day amanda blake Nothing too serious, you know, R and R. He kept remembering the night he spent in the hospital with the blond girl. Moreover he was beginning to feel paternal toward the young American. Sometimes he felt it was them against all the rest. Across the terrace from them, two CIA men stood by the railing with their suit jackets off, old-fashioned shoulder holsters strapped across their white shirts. Behind them, Coastown stretched out like a giant, glittering carnival. It would have been far more dignified to have reacted with cool unconcern, instead of shouting like that and then letting herself be won over by a simple smile. He caught her trying to slip out of the office unnoticed, and took her arm in a firm grasp. I like to keep a close eye on expenditure and select my investments very carefully.

How to Find the Angle Between Two Vectors: 12 Steps

The toe of her boot accidentally kicked the old, battered rocker on the porch. She muttered at the semi-darkness and dim porch light. The cluttered little porch grew deadly in the dark, and she felt grateful her mother left the light on for them. These little touches made her feel warm and happy. She hoped things would never change. Josh could be dressed as a sheikh. You have spent a lot of time in deserts. Still, she had better stay in her role, thought Bella, snuggling perversely into Josh and risking a kiss on his jaw. While I blessed my hand over his head in feigned forgiveness, I kicked him so hard in the balls that he writhed on the ground for a good five minutes. The brute repeated his evil deed the next day. In her eighth month, the Egyptian doctor kicked Cinfa down the stairs of their home.

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Release of this information garnered Diane an other mountain of mail wanting access to the mummy. The analysis on his kidney tumor showed it to be be nign. The mines were usually the only work to be had. They crowded into the same kind of squalid shantytown visible just outside every South African city. Precisely the kind of hardware the Cubans are using in Namibia. Is anyone else getting into the act. He was uncomfortable with ambiguities and liked to have everything he presented tied up in a nice, neat, typed package. He swiveled in his seat and took a folder bordered with red and white stripes from a silent aide. And too many foreign governments too willing to help the arms dealers keep a lid on their activities.

  • Defining the angle between vectors. Defining a plane in R3 with a point and normal vector. Cross product introduction. Proof: Relationship between cross product and sin of angle. So thats what tells us that these two things are equal or these two things are equal. So weve proven to ourselves that order doesnt matter when you take the dot
  • Direct way of computing clockwise angle between 2 vectors
  • The angle between two planes from the vectors dot product is: Plane - P 1: Plane - P 2: Plane - P 3: Condition for a single point of intersection is: rank R c = 3 rank R d = 3: Intersection point is given by Cramers rule: /// In vector
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You might as well have pushed her off yourself. As it was she only just spotted you in time. We could all have spent the night driving around in the dark looking for you. Anyone with the most basic knowledge of mechanics could have fixed it. rear trailing arm mounts frame repair trailblazer As in axis/angle representation, can use unit length quaternion for orientation: Represents a set of vectors forming a hypersurface of 4D hypersphere of radius 1 Hypersurface is a 3D volume in 4D space, but think of it as the same idea of a 2D surface on a 3D sphere q=s2+q 1 2+q 2 2+q 3 2=1Angle Between Two Vectors Calculator to find the angle between two vector components. The concept of the vector angle is used to describe the angle difference of physical quantities which have a magnitude and a direction associated with them. The vector angle is calculated from the endpoint of the first line to the endpoint of the second line. saitex logo Then her gaze dropped and she saw what he was holding in his hands. He could hardly have made it clearer that last night had indeed been a pretence as far as he was concerned. She turned her face away on the pillow. The whole day had taken on a nightmarish atmosphere. Over breakfast Mal had behaved as if absolutely nothing had happened between them. should i block my ex quora We took care of the family and we took care of the house. And then she bent over, gave Trixxi a pat, and was gone, watched by a loving pair of Labrador eyes. She had a glass of sherry on the table beside her, and she and Florence were settling in to watch the first episode of a new serial on television, involving the upstairs and downstairs lives of an aristocratic old family in a magnificent stately home. Certainly one of the most eventful. I should get out and about more.

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Finding The Angle Between Two Vectors - Calculus 3 - YouTube

And that area around it with all the narrow alleyways. Two since I heard from them, when there were some papers to sign. One of the boys who left earlier this year, poor misguided soul, is trying to get back at his wife by making her choose between her life here and her children. A very difficult time all around," she repeated. radarr rename existing files We rode the next few miles with him slumped forward against me, dreadfully uncomfortable for me but easier on him. However, when he began to jerk about behind me I was forced to relinquish the reins to Mahmoud and be led, riding half-doubled over and with both arms stretched behind me to keep Holmes from tumbling to the ground. At about this time our guide turned calmly into another road and, without acknowledgement from either side, rode away. A few minutes later Ali turned to check on us, then kicked his mare into a gallop and left us trotting along in a cloud of dust. Do you know how steep that cliff is. I assume you were asking me about the degree of climbing dif ficulty. He was about fifty feet down on the face of the cliff, coming after me. I was twenty feet away from him at the time. We were almost five hundred feet up on the side of a cliff.

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It seemed this fiend had seen plenty of battle. She ambled closer and I shielded my eyes and tried to sit up. You and I are merely scapegoats. The hand resting on my shoulder tensed, digging steely fingers into me. Yet when I discovered that her parting shot had been to let herself into our bathroom and help herself to shampoo and bath oil, her malice felt like sandpaper against sunburnt skin. Certainly, I was not Fanny Savage, wife, mother, wine expert and business woman which, once, had been my ambition. He was in his best grey suit and sported a light tan.


It was all very well to talk about scratching an itch, but, once you had given in to the need, it was almost impossible to stop. It was madness to think that she could sleep with him for three weeks and then calmly walk away. PLANES AND HYPERPLANES 5 Angle Between Planes Two planes that intersect form an angle, sometimes called a dihedral a Figure11:The angle between two planes is the same as the angle betweenAngle Between Two 3D Vectors Below are given the definition of the dot product (1), the dot product in terms of the components (2) and the angle between the vectors (3) which will be used below to solve questions related to finding angles between two vectors. doom weapon sounds When I went to get the car, you went out the back and came around the house. You were scraping off a stinger- you shoved it into one of the beehives. The third hive from the end is making queen cells at a tremendous rate, so I thought I might give them something else to think about. But do you mean to tell me that you anticipated. We are merely talking about likelihoods," he said with asperity, and placed the envelope in a breast pocket. The exquisite juxtaposition of ladylike threats and backhanded compliments proved irresistible. I met the colonel at my club, and they invited me out to their country place for a weekend. I kept my face without expression when Holmes turned to speak to me.

Still ignoring the two deputies, who were now standing on either side of her, she capped the bottles, tossed them into the trunk, ran her fingers through her shaggy hair to comb it roughly into place, and brought her right foot up to the bumper to untie her shoe. Only now did one of the young men speak, the one on her left. But do you mind telling us what you were doing. Feb 25, 2011The solid angle of orthogonal basis in 4D space must be α = 2 ⋅ π2 ⋅ R3 24 = π2R3 8. Here we can see a π2 factor. Does this mean that the 4D solid angle formula contains multiplication of two arctan there is one arctan in this formula? I think yes, but still have some difficulties with such formula inference. fcpl catalog Her eyes drifted shut, her face as soft as if she were sleeping, as trouble free. It made him feel good, manly, satisfied. She was his now, whether she admitted it or not. He could feel it in her surrender. Olivia began to shake her foot under the table. She was not accustomed to sitting still. During the war, it was the Whites, but they moved out of town before the armistice. The next family, the Carters, were there the longest.

When our tanker program is complete we can reduce that to between three and six months. Also our new bacteriological herbicide will be far more efficient. These organisms are biologically alive as distinct from chemically dead, so they reproduce themselves and actually increase their effectiveness from the moment of dispersal. A civilian member of the scientific liaison team, he was in here in the Prime Situation Center for precisely this purpose. The second factor is the complexity of the biosphere and the interaction of its various components: oceans, atmosphere, landmass, living organisms, and so on. Interpretation of the figures, as Colonel Madden has mentioned, varies a great deal. Use the calculator of Magnitude and Direction to Answer the Questions. Use the calculator to find the direction of the vectors u = < - 2 , 3 > and v = < - 4 , 6 >. Why are they equal? Find the direction of the vectors u = < 2 , 5 > and v = < - 2 , - 5 >. Why is the difference between the two directions equal to 180°?Two three-dimensional vectors a and b originate from the same coordinates area=( a 1 , a 2 , a 3 ) and b= ( b 1 , b 2 , b 3 ).Determine the angle between those vectors andhow should the how to screen share videos on zoom without lag I had contemplated marriage, had even played with the idea of suggesting it, but until he had himself referred so sarcastically to the marriage bed, I had never actually pictured him in those terms. That had bought me some breathing space, albeit brief and at a high price, but now I had it, I could not think what to do. Something would have to be done, that was obvious. I had not asked for this intolerable awareness, I did not want it, and I should have given a great deal to have it taken from me, but it had come, and I was in its grip. It was a small establishment with the cheerless and misleading name of the Vicissitude, but it did not allow its right-thinking feminist policies to interfere with the amount of hot water in its pipes or the quality of food that came from its kitchen. Amusing to think that I have spent twelve years of my life studying the patterns of transformation in this laboratory of mine, and you should come here and tell me something that is plain before my eyes. Your hidden talent, indeed, to see what is needed. I must remember to thank Steven.

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Colin wondered what the chief wanted. It was bound to be about the three murders, but how could he help. Behind the eight ball is what he is. Walking toward the Paradise, Mark caught up to him. Separate terms in each vector with a comma ",". The number of terms must be equal for all vectors. Vectors may contain integers and decimals, but not fractions, functions, or variables. About Dot Products. In linear algebra, a dot product is the result of multiplying the individual numerical values in two or more vectors.This vector addition calculator can add up to 10 vectors at once. DIRECTION must be entered in degrees, increasing counterclockwise. In rather unscientific terminology, a vector pointing directly to the right has a direction of zero degrees. A vector pointing straight up has an angle of 90 degrees. ineos operations And next year, or sometime soon, the director, top of the heap. She was at Scripps, taking her Ph. His confidence did, in fact, ease some of the frustration dragging at her heart. Equally clearly, none of those close to him had made the slightest push to influence his high-handed ways. The determination to make him change his attitude, at least with respect to her, grew with every short stride her meek chestnut took.

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Dot Product Calculator - Calculator Soup - Online Calculators

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  • ofVec2f | openFrameworks
  • Vector Dot Product Calculator - [100% Free] - Calculators.io
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  • The angle between the two displacements is 90 degrees so you can calculate the magnitude of the resultant displacement by using pythagoras. If you are comfortable with it you can also use the angle you just calculated and sine or cosine. With the magnitude of the displacement vector you can calculate the eventual time difference.

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All senses constantly alert, they dined, then, pickets organized, retired. All of them felt as if they were incessantly looking over their shoulders. He spoke quietly, so they had to listen hard to hear his wisdom. Have we not tried and tried, only to come away with our noses bloodied. No-we need a new plan, a better tactic. rent a masseur Her body was so near now that the guard of her dagger was digging into my thigh. Everything else lay in darkness, which meant that it was easy to melt into the shadow provided by a dilapidated wall that gave onto an abandoned garden. We positioned ourselves where we could see the door and window of the tavern. At the other end of the street, in the neighboring gloom of Calle de Tudescos, we could see a few ladies of the night casting their bait-with little success. Now and then, men, singly or in groups, would enter or leave the inn. Yesterday had been a dark dream plagued by a neurotic fantasy, created by pressures and anxieties and fed by jet lag, her personal history, and an accumulation of sleepless nights. Today she would start afresh, and give herself a chance to see this branch of Change with clear eyes.

Vectors Calculator

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Some curious aversion to the ship itself. Furthermore, it is the place where, at the age of fourteen, you experienced the horrendous crash that killed your mother, your father, your brother, and nearly you. It would be decidedly odd if you were not fearful. What concerns me is that your degree of apprehension seems excessive. Those dreams, whatever their message, clearly spring from powerful roots. ai shoujo dhh setting Mar 18, 2013The vector_difference function is used to calculate the difference of two vectors online. Calculating the norm of a vector: vector_norm. The vector calculator allows the calculation of the norm of a vector online. Calculating the sum of two vectors: vector_sum. The vector calculator allows the calculation of the sum of two vectors online. sobredosis de fluoxetina muerte For Edmonton, though, Catholics were faceless and anonymous - barbarians amassing at the gate to sack Protestant civilisation - and therefore could be subjected to any degree of inhumanity in the name of a nobler cause. Closeted in his English home, and buoyed by a formless hatred, Edmonton had overseen a chain of events that had led to many deaths. But it was pointless to expect him to feel guilt for what he had done. Catholic emancipation passed through both Houses. But he was perspiring like a hog. While you were preparing for bed, I found it under one of the pillows and disarmed it. delta math answers probability A buzzing as her hair was shaved, a cold wash against the scalp, and eventually a mask on her face. Cops like pillars, doctors like whips, these sensations washed over her while she lay stunned and unmoving, imprinting their textures on her battered brain, to appear in later life - never while she was conscious, but as dream images: fellow cops who smelled of dust, a nurse covered with luscious warm fur, words that tasted like broken glass.




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Vector space - Wikipedia

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  • How to Use the Angle Between Two Vectors Calculator? The procedure to use the angle between two vectors calculator is as follows: Step 1: Enter the coefficient of the components of the vector in the input field. Step 2: Now click the button “Find Angle Between A and B” to get the result. Step 3: Finally, the angle between the two vectors will be displayed in the output field. What is Meant by Angle Between Two Vectors?
  • Angle Between Two Vectors Calculator
  • Jul 23, 2018
  • How to find angle between two vectors with magnitudes?

The door to forever opened before us as we walked out of the chapel as husband and wife. The photographer kept us occupied for about another half hour and then we finally joined the celebration. Everything always came back to that. He turned to his typewriter, stared at the piece of paper still there from the night before. He reached for the bound volume of the Gazette seventy-five years before for the week of May 26th. One of his duties was to do the "Looking Back" column. But that was the way it was on a small weekly: You did all kinds of stuff no matter what your title was. Colin knew it was something you could do in half an hour, but he liked reading all the ads, the sports and the real estate section. After a few moments, his cigarette bobbing dangerously close to his black beard, he paused in his carving and raised his eyes to Holmes. Mycroft is something by way of an expert on keeping things to himself. Mycroft thought we could as well make ourselves useful as hide in a cave somewhere. We do not have time to set our steps by yours. An hour in the bazaar to supply the portions of our costume the boat could not provide, and we are ready.

Both are beautiful, full and healthy. She decided to tell him to mind his own business when he turned to look at Matt. How did she explain that she refused to admit her father had died. If she could keep everything the same, he would still be there. A part of him… Her eyes burned with tears, but she kept them at bay. It would make her feel better to keep everything the same. And this from a woman, to whom he could hardly retort in kind. The ground here was lower and spotted with bog plants, with a small lake glittering below. We worked our way up the end of the wet area, where a tiny stream trickled clear and cold over a tumble of water-rounded stones. I slipped the stones into my pocket and took up my gun. The two Gerard boys, both of them now armed and Marsh behind one of them looking on, had great success. Unbidden, the memory of his odd behaviour in the hall rose to taunt her. A frown chased the smile from her eyes.

I could feel safe with Jonathan, I knew. Hunched and sullen, I sat in the departure lounge at Heathrow, nursing a beaker of tea. I wanted to be at home, in bed, soon to begin my nice, safe routine. I woke up at half past six and made myself a cup of tea. There was still some open range left, in spite of the encroaching sodbusters and squatters. There was a wild man locked up inside him. I mean, never waste a crisis, right. It was from outside, down on Broadway-a loud metal thump, followed by tires shrieking and then a long, wailing scream. It was a young woman, her black skirt completely torn on one side, her white shirt covered in blood. I knelt beside her and then reared back as I saw the short black hair and the face framed by silver hoop earrings. There were cigarette holes along her collarbone and two star-shaped, point-blank bullet wounds in her right and left cheeks.

Some have already left for the rendezvous point. Broken Covenant must be called off. Despite the crowding, they listened quietly to their teacher as he ran through the alphabet again. Dulcie looked up from her work, nodded, and dropped her head again. She thanked Carla, said good-bye to her and to the child, and went back out onto the street, the bell tinkling behind her. On the doorstep Carla, bent over the lock, felt Dulcie tug at her sleeve. She had a truly ugly hat on, an orange knit thing that was all lumpy. Blue leggings, red high-top athletic shoes. The camera was a thirty-five millimeter with a long lens, kind of beat-up-looking, and she was running around telling people where to stand. Well, sort of unstructured," Kate said.

Neither of them thought the butler was lying. A bevy of stableboys rushed past, racing to take the high-stepping horses in charge. The master had to half carry her up to the house. Just beyond the arch, he halted and looked about. Their Majesties had appeared on the other side of the balustrade, at the top of the main staircase. I watched as she walked away and as she turned once to smile at me. Then he remembered the Angolans and Cubans were in the act. He had the unpleasant feeling he was about to find out.