SEMESTER 1
ASSIGNMENT 02
Fixed closing date: 06 March 2017
Unique assignment number: 892294
QUESTION 1
If each component of a non-zero vector in R3 is doubled then the length of that vector is doubled.
Prove this statement. (5)
QUESTION 2
Suppose that u and v are two vectors such that jjujj = 2; jjvjj = 1 and u · v = 1. Find the angle
between u and v in radians. (5)
QUESTION 3
(a) Suppose the relationship Projau = Projav is true for some vectors a; u and w:
(i) Verify that u · v = v · a. (5)
(ii) Provide a counte 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) (u × v) × w 6= u × (v × w) sometimes and that (5)
(ii) if u 6= 0; u × v = u × w 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 = (-2; 0; 4) ; v = (3; -1; 6) and w = (2; -5; -5) : Compute
(i) the area of the parallelogram bounded by u and v (5)
(ii) the equation of the plane parallel to v and w passing through the tip of u. (5)
QUESTION 6
Let z1 = x + iy and z2 = a + ib with z1 = z2: Prove that
(i) x2 - a2 = (b - y) (b + y) and (5)
(ii) The arguments of z1 and z2 differ by a multiple of 2p (5)