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Quiz Chapter 8: Vectors

10 questions · Form 4 Additional Mathematics Bab 8: Vectors

10 questions · plays in fullscreen

Full Question List & Answer Key

Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.

1. Vector a has magnitude 10 units making an angle with the positive x-axis such that cos θ = 0.6 and sin θ = 0.8 in the first quadrant. Express vector a in Cartesian component form.

  1. 6i + 8j
  2. 8i + 6j
  3. 0.6i + 0.8j
  4. 6i - 8j
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Answer: A

Vector a = |a|cosθ i + |a|sinθ j = 10(0.6)i + 10(0.8)j = 6i + 8j.

2. Given a vector a = 6i - 8j. Calculate the unit vector in the direction of a.

  1. (35)i - (45)j
  2. (610)i + (810)j
  3. -(35)i + (45)j
  4. (45)i - (35)j
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Answer: A

|a| = √62 + (-8)2 = √36 + 64 = √100 = 10. The unit vector = a / |a| = 6i - 8j10 = (35)i - (45)j.

3. If vector AB = ~a + 3~b and vector CD = 4~a + 12~b, which statement correctly describes the relationship between AB and CD?

  1. AB is perpendicular to CD
  2. AB is parallel to CD and CD = 4AB
  3. AB and CD are unit vectors
  4. AB is parallel to CD and AB = 4CD
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Answer: B

CD = 4~a + 12~b = 4(~a + 3~b) = 4AB. Since CD = k*AB with k = 4 (a non-zero scalar), AB is parallel to CD and CD = 4AB.

4. Given two non-zero vectors u and v such that u = (k - 3)v. If u and v are parallel, which statement regarding k is correct?

  1. k can be any real number except 3
  2. k must be greater than 0
  3. k must equal 0
  4. k = 3
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Answer: A

For non-zero vectors to be parallel, u = m*v where scalar m ≠ 0. Here m = k - 3, so k - 3 ≠ 0, which means k can be any real number except 3.

5. In a triangle ABC, vector AB = u and vector AC = v. M is the midpoint of BC. Express vector AM in terms of u and v.

  1. 12 (u + v)
  2. u + 12 v
  3. 12 (u - v)
  4. v - 12 u
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Answer: A

BC = AC - AB = v - u. BM = 12 BC = 12 (v - u). AM = AB + BM = u + 12(v - u) = 12 u + 12 v = 12 (u + v).

6. Given OP = 4i - 3j and OQ = (k + 1)i + 9j. If OP is parallel to OQ, find the magnitude of OQ.

  1. 15
  2. 20
  3. 25
  4. 18
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Answer: A

Since OP \parallel OQ, OQ = m*OP => (k+1)i + 9j = m(4i - 3j). From j-components: 9 = -3m => m = -3. Thus OQ = -3(4i - 3j) = -12i + 9j. Magnitude |OQ| = √(-12)2 + 92 = √144 + 81 = √225 = 15.

7. Vectors a and b are non-zero and non-parallel. If (h - 2)a + (k + 5)b = 0, find the values of h and k.

  1. h = 2, k = -5
  2. h = -2, k = 5
  3. h = 2, k = 5
  4. h = 0, k = 0
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Answer: A

Since a and b are non-zero and non-parallel, (h - 2)a + (k + 5)b = 0 holds if and only if both coefficients are zero: h - 2 = 0 => h = 2, and k + 5 = 0 => k = -5.

8. The position vectors of points A and B relative to origin O are ~a = 2i + 7j and ~b = 5i - 2j respectively. Express AB in column vector form.

  1. (-3, 9)^T
  2. (7, 5)^T
  3. (3, -9)^T
  4. (-7, -5)^T
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Answer: C

AB = OB - OA = (5i - 2j) - (2i + 7j) = (5-2)i + (-2-7)j = 3i - 9j, which in column vector form is (3, -9)^T.

9. Given 3a - 2b = c. If a = 2i + j and c = 4i - 5j, find vector b.

  1. i + 4j
  2. -i - 4j
  3. 2i - 8j
  4. i - 4j
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Answer: A

3a - c = 2b => 2b = 3(2i + j) - (4i - 5j) = (6i + 3j) - (4i - 5j) = 2i + 8j. Dividing by 2 gives b = i + 4j.

10. Given that vector u = (p + 2)i + 12j is parallel to vector v = 3i + 4j. Determine the value of p.

  1. 7
  2. 9
  3. 5
  4. 11
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Answer: A

Since u \parallel v, u = k*v => (p + 2)i + 12j = k(3i + 4j). From j-components: 12 = 4k => k = 3. From i-components: p + 2 = 3k = 3(3) = 9 => p = 7.

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