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Verify the following:(i) $(0,7,-10),(1,6,-6)$ and $(4,9,-6)$ are the vertices of an isosceles triangle.(ii) $(0,7,10),(-1,6,6)$ and $(-4,9,6)$ are the vertices of a right angled triangle.(iii) $(-1,2,1),(1,-2,5),(4,-7,8)$ and $(2,-3,4)$ are the vertices of a parallelogram.
Calculus 3
Chapter 12
Introduction to Three Dimensional Geometry
Section 2
Coordinate Axes and Coordinate Planes in Three Dimensional Space
Vectors
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in part A. We want to prove that the straight points. The vortices of a source list triangle. So we need to compute the nerves. The victory be the man's of the vector A. C. And the finance of the vector Beastie the ninth of the vector A. B. Equals the square road child. One square pass miners. One square plus four square Richard Curls the Square Road of 18. The length of a vector A. C. He calls the square root of four squared plus two squared plus four squared Which equals the square root of 36 regional six announce of the vector Bc equals the square root of three square plus three square plus zero square ventricles The Square Root of 18. So we know that the nails of A. B. Details the nance obesity enhance. It is at softly as Crandall. And per big. We want to prove these three points. The verses of a right and daughter triangle. We can write the vectors A. B retreat curls minus one. My nurse one my nurse for. And the vector A. C. Asked miners. Four. Two minor school vector Bc. Because minus three scream zero adult products of A. B. And B. C. And curls -1 times -3 Plus -1 times three Plus -4 times zero which recall zero in the hands. A. B. Yes, perpendicular to basic. Thus the triangle is a right angled triangle in percy Iran to prove that there's four points. Uh verses of paranoia to win. Still we need to compute A vector A B. Director A. D. Director D. C. And the Rector Beastie. As you can see this is a parallelogram. So a b. Eco's to minors. four. Well, AD. Eco's 3 -5 3 DC Equals 2 -4, 4, And BC Echoes Scree -5 3. So the Rectory B. Because of record is sick, the vector A. D. Because of Rector Bc. And the hands A. B. Is parallel to D. C. And A. D. Is parallel to B. C. Hence these four points, the verses of the tyranny anagram.
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