Linear Combination and Span of Vector Space and Sub Space:2. Show that the xy plane W = (x, y,0) in R³ is generated by:2(i)u = |2and1 (ii)uand3v =v =

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Answered: Linear Combination and Span of Vector…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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If L is a line in 2-space or 3-space that passes through the points A and B, then the distance from a point P to the line L is equal to the length of the component of the vector AP that is orthogonal to the vector AB P A В Use the method above to find the distance from the point P(-3,1,7) to the line through A(1,1,0) and B(-2,3, –4). NOTE: Enter the exact answer. Distance = 7.494825801
W= Let u = 4 8- and = 4 H 3 Find a vector w not in span {u, v}.
Let ri = xi+y1j+2ık, r2 = x2i+ Y2j + zzk and r3 = x3i+ Y3j+ z3k %3D be the position vectors of P points P¡(x1, Y1, 21), P2(x2, Y2, 2) P1 and P3(x3, Y3, 23). Find an equa- Ps Ts tion for the plane passing through P1, P2 and P3. See fig- ure on the right
Let r = <x, y, z>, a = <1, 3, −5>, and b = <5, −3, 1>. Show that (i) the vector equation (r − a) · b = 0 represents a plane, and find a point in that plane, a normal vector to that plane, and an equation of that plane; (ii) the vector equation (r − a) × b = 0 represents a line, and find a point on that line, a direction vector, and symmetric equations of that line; (iii) the vector equation (r − a) · (r − b) = 0 represents a sphere, and find its center, radius, and standard equation.
(b) Let P, be the plane in R which contains the line in R' with vector equation 6. X1 =| 5 + and the point (3, 2, 5). (This point is not on the line.) (i) A vector equation of P, is 6 X = 5 +s 4 +t 6. 6 (Use two points on the plane to find one of the direction vectors.) (ii) An equation in general form for P2 is x+ y+ To find this equation, you need a vector n = [a, b, c] which is orthogonal to the two direction vectors that you obtained in part (i).
Let u=i-3j+2k, v=i+j, and w=2i+2j−4k be vectors in 3-space. Find: (a)u+1 (b) unit vector in the direction of w (c) u.(v+w) (d) the angle between u and y (e) the direction cosines of the vector u (f) the orthogonal projection of v onto w (g) uxv (h) u.(vxw)
A plane P contains two intersecting lines given by Ti (1,1,0) +t(1,-1,2) = T2 (2,0,2) + s(-1,1,0). = (a) Find the equation of the plane P that contains these two lines. (b) Find the vector projection of = (6,2, 1) onto the plane P.
d-distance between vectors

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Linear Combination and Span of Vector Space and Sub Space: 2. Show that the xy plane W = (x, y,0) in R³ is generated by: (i)u and 1 (ii)u = and