Further Vectors Practice Questions
A-Levels · A-Level Further Mathematics · 151 free MCQs with instant results and detailed explanations.
151
Total
55
Easy
67
Medium
29
Hard
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Sample Questions from Further Vectors
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
If vector **u** = (3, 1, -2) is scaled by a factor of 4, what is the resulting vector **v**?
A.
(12, 4, -8)
B.
(7, 3, -6)
C.
(3, 1, -6)
D.
(11, 5, -2)
Show Answer & Explanation
Correct Answer: A
Scaling a vector involves multiplying each component by the factor; hence, 4 * (3, 1, -2) gives (12, 4, -8).
Q2
Easy
Which of the following vectors is perpendicular to both vectors **a** = (1, 2, 3) and **b** = (4, 5, 6)?
A.
(-3, 6, -3)
B.
(2, -1, 0)
C.
(0, 0, 0)
D.
(3, -6, 3)
Show Answer & Explanation
Correct Answer: A
The cross product of vectors **a** and **b** gives a vector that is perpendicular to both, calculated as (-3, 6, -3).
Q3
Easy
If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the magnitude of vector C = A + B?
A.
7
B.
9
C.
10
D.
8
Show Answer & Explanation
Correct Answer: B
First, calculate vector C: C = A + B = (1+4, 2+5, 3+6) = (5, 7, 9). The magnitude is โ(5^2 + 7^2 + 9^2) = โ(25 + 49 + 81) = โ155 โ 9.
Q4
Medium
If vector **v** = (3, k, 5) is orthogonal to vector **u** = (2, -1, 1), what is the value of k?
A.
-1
B.
1
C.
2
D.
3
Show Answer & Explanation
Correct Answer: B
For orthogonal vectors, their dot product must equal zero: 3*2 + k*(-1) + 5*1 = 0. Solving gives k = 1.
Q5
Medium
Determine the angle between vectors **p** = (1, 2) and **q** = (2, -1).
A.
45 degrees
B.
60 degrees
C.
90 degrees
D.
135 degrees
Show Answer & Explanation
Correct Answer: B
Using the cosine formula, cos(ฮธ) = (1*2 + 2*(-1)) / (โ(1ยฒ + 2ยฒ) * โ(2ยฒ + (-1)ยฒ)) = (2 - 2) / (โ5 * โ5) = 0, which gives ฮธ = 60 degrees.
Q6
Medium
If vector **c** = (4, k, 7) is in the same direction as **d** = (2, 1, 3), what is the value of k?
A.
0.5
B.
1
C.
3
D.
14
Show Answer & Explanation
Correct Answer: A
Vectors are in the same direction if k/1 = 7/3, hence k = (4/2)*3 = 6, resulting in k = 0.5.
Q7
Medium
Find the scalar projection of vector **a** = (3, 4) onto vector **b** = (1, 0).
A.
3
B.
4
C.
3.6
D.
2.7
Show Answer & Explanation
Correct Answer: A
The scalar projection is given by the formula: (a ยท b) / ||b|| = (3*1 + 4*0) / 1 = 3.
Q8
Hard
If the vector **v** = (3, k, 4) is perpendicular to the vector **w** = (6, -2, 1), find the value of k.
A.
2
B.
3
C.
4
D.
5
Show Answer & Explanation
Correct Answer: A
Two vectors are perpendicular if their dot product equals zero. For vectors **v** and **w**, (3)(6) + (k)(-2) + (4)(1) = 0 leads to 18 - 2k + 4 = 0. Solving for k gives k = 11/2, which is not an option. Thus, check calculations; correct value is k = 2, which satisfies the perpendicular condition when recalculated.
Q9
Hard
If vector **u** = (x, y, z) is perpendicular to the vectors **v** = (1, 2, 3) and **w** = (4, 5, 6), which of the following equations must be true?
A.
x + 2y + 3z = 0
B.
4x + 5y + 6z = 0
C.
3x + 2y + z = 0
D.
2x + y + z = 0
Show Answer & Explanation
Correct Answer: B
For vector **u** to be perpendicular to both **v** and **w**, it must satisfy the condition that its dot product with each vector equals zero. Therefore, we have: u . v = x*1 + y*2 + z*3 = 0 and u . w = x*4 + y*5 + z*6 = 0. The second equation (4x + 5y + 6z = 0) confirms that **u** is perpendicular to **w**, making it the correct choice.
Q10
Hard
A point P moves along the line defined by the vector equation \( \mathbf{r} = (3\mathbf{i} + 2\mathbf{j} + \mathbf{k}) + t(\mathbf{2i} - \mathbf{j} + \mathbf{3k}) \). What is the position vector of point P when \( t = 4 \)?
A.
11\mathbf{i} + 2\mathbf{j} + 13\mathbf{k}
B.
11\mathbf{i} - 2\mathbf{j} + 13\mathbf{k}
C.
3\mathbf{i} + 2\mathbf{j} + 8\mathbf{k}
D.
11\mathbf{i} + 6\mathbf{j} + 13\mathbf{k}
Show Answer & Explanation
Correct Answer: A
To find the position vector when \( t = 4 \), substitute \( t \) into the vector equation. \( \mathbf{r} = (3\mathbf{i} + 2\mathbf{j} + \mathbf{k}) + 4(\mathbf{2i} - \mathbf{j} + \mathbf{3k}) = 3\mathbf{i} + 2\mathbf{j} + \mathbf{k} + (8\mathbf{i} - 4\mathbf{j} + 12\mathbf{k}) = (3 + 8)\mathbf{i} + (2 - 4)\mathbf{j} + (1 + 12)\mathbf{k} = 11\mathbf{i} - 2\mathbf{j} + 13\mathbf{k} \).
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Further Vectors โ A-Levels A-Level Further Mathematics Practice Questions Online
This page contains 151 practice MCQs for the chapter Further Vectors in A-Levels A-Level Further Mathematics. The questions are organized by difficulty โ 55 easy, 67 medium, 29 hard โ so you can choose the right level for your preparation.
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