Precalculus Practice Test: Vectors
- Vectors
- 10 questions
- 15 minutes
These 10 questions cover component form, vector addition and scalar multiplication, magnitude, unit vectors, the dot product and angles between vectors. Two nonzero vectors are perpendicular exactly when their dot product is 0.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
Let u = ⟨3, −2⟩ and v = ⟨−1, 5⟩. What is 2u − v?
Correct answer: D
First 2u = ⟨6, −4⟩, then subtract v: ⟨6 − (−1), −4 − 5⟩ = ⟨7, −9⟩. The vector ⟨5, 1⟩ is 2u + v.
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Question 2
What is the magnitude of the vector ⟨−6, 8⟩?
Correct answer: A
The magnitude is (−6)2 + 82 = 100 = 10. The value 100 forgets the square root, and 14 adds the absolute values of the components.
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Question 3
What is the unit vector in the direction of ⟨5, −12⟩?
Correct answer: C
The magnitude is 25 + 144 = 13, and dividing each component by 13 gives a vector of length 1 in the same direction. The vector with −5/13 and 12/13 points the opposite way.
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Question 4
What is the dot product ⟨2, 3⟩ · ⟨4, −1⟩?
Correct answer: B
Multiply matching components and add: 2(4) + 3(−1) = 8 − 3 = 5. A dot product is a number, so ⟨8, −3⟩, which only lists the products, is not the answer.
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Question 5
What is the angle between the vectors ⟨1, 0⟩ and ⟨1, 3⟩?
Correct answer: D
Use cos θ = (u · v)/(|u||v|) = 1/(1 · 2) = 1/2, so θ = 60°. The angle 30° is the angle the second vector makes with the y-axis.
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Question 6
For what value of k are ⟨4, k⟩ and ⟨3, −6⟩ perpendicular?
Correct answer: A
Perpendicular vectors have a dot product of 0: 12 − 6k = 0, so k = 2. The value −2 drops the negative sign of −6.
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Question 7
What is the component form of the vector from P(2, −1) to Q(−3, 4)?
Correct answer: C
Subtract the initial point from the terminal point: ⟨−3 − 2, 4 − (−1)⟩ = ⟨−5, 5⟩. The vector ⟨5, −5⟩ goes from Q to P, and ⟨−1, 3⟩ adds the coordinates.
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Question 8
A vector has magnitude 10 and direction angle 30° measured from the positive x-axis. What is its component form?
Correct answer: B
The components are 10 cos 30° = 5√3 and 10 sin 30° = 5. The vector ⟨5, 5√3⟩ swaps sine and cosine, which gives a direction of 60°.
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Question 9
Two forces act on an object: 30 newtons pulling east and 40 newtons pulling north. What is the magnitude of the resultant force?
Correct answer: D
The forces are perpendicular, so the resultant has magnitude 302 + 402 = 2500 = 50 newtons. Adding the magnitudes to get 70 would only work if both forces pulled in the same direction.
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Question 10
What is the direction angle of the vector ⟨−3, 3⟩, measured counterclockwise from the positive x-axis?
Correct answer: A
The vector points up and to the left, into Quadrant II, with reference angle 45°, so the direction angle is 180° − 45° = 135°. A calculator gives tan−1(−1) = −45°, which points into Quadrant IV, the opposite side.
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