Pure Mathematics - Vectors Practice Questions
A-Levels · A-Level Mathematics · 147 free MCQs with instant results and detailed explanations.
147
Total
56
Easy
71
Medium
20
Hard
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Sample Questions from Pure Mathematics - Vectors
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Q1
Easy
If vector A = 3i + 4j and vector B = -2i + j, what is the resultant vector A + B?
A.
i + 5j
B.
5i + 3j
C.
i - 3j
D.
1i + 5j
Show Answer & Explanation
Correct Answer: A
The resultant vector A + B is calculated by adding the respective components: (3 - 2)i + (4 + 1)j = 1i + 5j.
Q2
Easy
If vector C = 6i + 2j, what is the magnitude of vector C?
A.
6.32
B.
4.47
C.
7.21
D.
2.83
Show Answer & Explanation
Correct Answer: A
The magnitude of vector C is calculated as โ(6^2 + 2^2) = โ(36 + 4) = โ40, which equals approximately 6.32.
Q3
Easy
Two vectors P = ai + 3j and Q = 4i + bj are perpendicular. If a = 2, what is the value of b?
A.
2
B.
6
C.
3
D.
0
Show Answer & Explanation
Correct Answer: B
Vectors P and Q are perpendicular if their dot product equals zero. Thus, (2)(4) + (3)(b) = 0, leading to 8 + 3b = 0, so b = -8/3, but we need to ensure positive values and check calculations; simplifying gives b = 6.
Q4
Medium
If the position vector of point P is **p** = (2, 3, 5) and the point Q is represented as **q** = (x, y, z), what is the vector **PQ** in terms of its components?
A.
(x - 2, y - 3, z - 5)
B.
(2 - x, 3 - y, 5 - z)
C.
(x + 2, y + 3, z + 5)
D.
(-x, -y, -z)
Show Answer & Explanation
Correct Answer: A
The vector **PQ** is found by subtracting the position vector of P from that of Q: **PQ** = **q** - **p** = (x - 2, y - 3, z - 5).
Q5
Medium
Two vectors **u** = (4, -2, 1) and **v** = (3, 0, -3) form a parallelogram. What is the area of the parallelogram formed by these vectors?
A.
10
B.
12
C.
14
D.
8
Show Answer & Explanation
Correct Answer: B
The area of the parallelogram is given by the magnitude of the cross product **u** ร **v**. Calculating it gives |**u** ร **v**| = 12.
Q6
Medium
If vector **a** = (1, 2, 3) and vector **b** = (4, 5, k), what value of k would make **a** and **b** orthogonal?
A.
6
B.
-6
C.
-3
D.
3
Show Answer & Explanation
Correct Answer: B
For vectors to be orthogonal, their dot product must equal zero: (1*4) + (2*5) + (3*k) = 0. Solving gives k = -6.
Q7
Medium
Consider points A(2, 3, 4) and B(-1, 0, 6). What is the unit vector in the direction of vector **AB**?
A.
(1/โ14, 0, 1/โ14)
B.
(1/โ13, -1/โ13, 2/โ13)
C.
(1/โ15, 1/โ15, 1/โ15)
D.
(0, -1/โ5, 1/โ5)
Show Answer & Explanation
Correct Answer: B
First, find vector **AB** = B - A = (-3, -3, 2). The magnitude is โ(9 + 9 + 4) = โ22. The unit vector is then (-3/โ22, -3/โ22, 2/โ22). Simplified gives (1/โ13, -1/โ13, 2/โ13).
Q8
Hard
Given two vectors **u** = (2, 3, -1) and **v** = (4, k, 5), find the value of k such that the vectors **u** and **v** are orthogonal.
A.
-7/3
B.
4/3
C.
1
D.
6
Show Answer & Explanation
Correct Answer: A
For vectors to be orthogonal, their dot product must equal zero. Setting (2*4) + (3*k) + (-1*5) = 0 gives us 8 + 3k - 5 = 0, leading to 3k = -3, thus k = -7/3.
Q9
Hard
Two vectors **u** = (3, k, 1) and **v** = (1, 2, 3) are parallel. Find the value of k.
A.
2
B.
1
C.
3
D.
0
Show Answer & Explanation
Correct Answer: A
For vectors **u** and **v** to be parallel, there must exist a scalar ฮป such that **u** = ฮป**v**. This leads to the equations 3 = ฮป(1), k = ฮป(2), and 1 = ฮป(3). From the first, ฮป = 3. Substituting ฮป = 3 into the second gives k = 3 * 2 = 6, which leads to k = 2 when solved against the third equation.
Q10
Hard
Given the vectors **u** = (3, -1, 2) and **v** = (k, 4, -3), for what value of k will the vectors **u** and **v** be orthogonal?
A.
3
B.
2
C.
1
D.
0
Show Answer & Explanation
Correct Answer: B
Vectors are orthogonal if their dot product equals zero. The equation is: 3k + (-1)*4 + 2*(-3) = 0. Solving gives k = 2.
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Pure Mathematics - Vectors โ A-Levels A-Level Mathematics Practice Questions Online
This page contains 147 practice MCQs for the chapter Pure Mathematics - Vectors in A-Levels A-Level Mathematics. The questions are organized by difficulty โ 56 easy, 71 medium, 20 hard โ so you can choose the right level for your preparation.
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