Number in brackets ( ) is the point value of the question.

1)

a) Find det (A)
b) Using det (A) and adj (A) find A ^{– 1} (5)

2) u = , v =

Find

a) 3u

b) u + v

c) v – u

d) 2u – 7v (4)

3) u = , v = If A is the angle between u and v, find cos A. (3)

4) u = 3i – 4j , and v = 4i + 3j. Prove u and v are orthogonal vectors. (2)

.

5) u = , v = find the value of b that makes u orthogonal to v. (3)

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6) u = , v = find w₁ and w₂ such that w₁ = proj_v u and w₂ = u – w₁. (4)

7) Prove that if v is a nonzero vector in R², then (4)

.

8) Find the perpendicular distance from P(2, 6) to the line 2x + y – 8 = 0. (3)

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9) Find the area of the parallelogram with consecutive vertices
at P(1, 3, –2), Q(2, 1, 4), R( – 3, 1, 6) using the cross product of 2 vectors. (4)

10) u = , v =
a) Find w such that w z both u and v.
b) Find x such that x z both u and v and Èx È = 1 (3)

11)

a) Write the parametric equations for the line l which passes through
P(2, – 1, 6) and Q(3, 1, – 2). (3)

b) Write the symmetric equations for the line through
P( – 1, – 1, 3) and is parallel to v = (3)

12)

a) Find an equation for the plane through P(2, 5, 1) with normal parallel
to the vector through Q( – 1, 2, 0) and R(2, – 6, 5). (3)

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b) Find an equation for the plane through P(1, 2, 1), Q( – 2, 3, – 1) and R(1, 0, 4). (4)

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c) Prove that the planes 2x + 3y – z = 3 and – 4x – 6y + 2z = 8 are parallel. (3)

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d) Find parametric equations for the line of intersection of
the planes 2x – y – z = 3 and x + 2y + 3z = 7. (4)

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e) Find the perpendicular distance from P(1, – 2, 3) to the plane 2x – y – z = 6. (3)

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f) Find all points of intersection of the plane 3x + 4y + 5z = 60 and
the line with parametric equations x = 5 + t, y = -2 + 2t, z = 7 – t (3)

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TOTAL (65)

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