Please help need full Answers to Question 1 - 6
QUESTION 1
If each component of a non-zero vector in R3 is trippled then the length of that vector is trippled.
Prove this statement. (5)
QUESTION 2
Suppose that u and v are two vectors such that IluII = 2, I M I = 1 and u • v = 1. Find the angle
between u and v in radians. (5)
QUESTION 3
(a) Suppose the relationship Proj.0 = Projo is true for some vectors a, u and w.
(i) Verify that a • a = p_ • a. (5)
(ii) Provide a counter example to show that u need not be equal to v in your example. (5)
QUESTION 4
Suppose u, v and w are vectors in R3. Show by means of a counter example that
(i) x x y2 u x (It x w) sometimes and that (5)
(ii) if y Q, axv=uxw. then u need not be equal to w. (5)
[Hint: Use i = (1, 0, 0) , j = (0,1, 0) and k = (0, 0,1)]
QUESTION 5
Let u = (1, 0, 2) , y = (0, —1, 2) and w = (2, 1, 0) . Compute
(i) the area of the parallelogram bounded by u and v (5)
(ii) the equation of the plane parallel to v and tel passing through the tip of u. (5)
QUESTION 6
Let z1 = x + iy and z2 = a + ib with zi = z2. Prove that
(i) 3x2 — 3a2 = (b — y)(b + y) and (5)
(ii) The arguments of z1 and z2 differ by a multiple of 2/r (5)