Vector Multiplication — NEET Physics MCQs with Solutions
Free NEET Physics Vector Multiplication MCQs with step-by-step solutions (35 questions). Part of Vectors. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Vector Multiplication · easy · numerical
If $\vec A = 2\hat{i} + 3\hat{j}$ and $\vec B = \hat{i} - \hat{j}$, then $\vec A \cdot \vec B$ is
A. $1$
B. $-1$ ✓ Correct
C. $5$
D. $-5$
Solution: $\vec A\cdot\vec B = (2)(1) + (3)(-1) = 2 - 3 = -1$.
Q2 — Vector Multiplication · medium · theory
The angle between $\vec A = \hat{i} + \hat{j}$ and $\vec B = \hat{i}$ is
A. $90^\circ$
B. $60^\circ$
C. $45^\circ$ ✓ Correct
D. $30^\circ$
Solution: $\cos\theta = \frac{\vec A\cdot\vec B}{AB} = \frac{1}{\sqrt{2}\cdot 1} \Rightarrow \theta = 45^\circ$.
Q3 — Vector Multiplication · medium · numerical
The value of $\lambda$ for which $\vec A = 2\hat{i} + \lambda\hat{j}$ is perpendicular to $\vec B = 3\hat{i} - 2\hat{j}$ is
A. $-3$
B. $6$
C. $3$ ✓ Correct
D. $\frac{2}{3}$
Solution: Perpendicular $\Rightarrow \vec A\cdot\vec B = 0$: $6 - 2\lambda = 0 \Rightarrow \lambda = 3$.
Q4 — Vector Multiplication · medium · numerical
The projection of $\vec A = \hat{i} + 2\hat{j} + \hat{k}$ on $\vec B = 2\hat{i} + \hat{j} + 2\hat{k}$ is
A. $2$ ✓ Correct
B. $3$
C. $\sqrt{6}$
D. $6$
Solution: Projection $= \frac{\vec A\cdot\vec B}{|\vec B|} = \frac{2+2+2}{\sqrt{4+1+4}} = \frac{6}{3} = 2$.
Q5 — Vector Multiplication · easy · numerical
A force $\vec F = (3\hat{i} + 4\hat{j})$ N displaces a body by $\vec d = (2\hat{i} + \hat{j})$ m. The work done is
A. $10$ J ✓ Correct
B. $11$ J
C. $14$ J
D. $5$ J
Solution: $W = \vec F\cdot\vec d = (3)(2) + (4)(1) = 10$ J.
Q6 — Vector Multiplication · easy · numerical
$\hat{i}\cdot(\hat{j}\times\hat{k})$ equals
A. $0$
B. $\hat{i}$
C. $-1$
D. $1$ ✓ Correct
Solution: $\hat{j}\times\hat{k} = \hat{i}$, and $\hat{i}\cdot\hat{i} = 1$.
Q7 — Vector Multiplication · medium · theory
$\hat{i}\times\hat{j}$ equals
A. $-\hat{k}$
B. $\hat{j}$
C. $\hat{k}$ ✓ Correct
D. $0$
Solution: By the right-hand rule for the cyclic order $x\to y\to z$: $\hat{i}\times\hat{j} = \hat{k}$.
Q8 — Vector Multiplication · easy · numerical
If $|\vec A| = 4$, $|\vec B| = 3$ and the angle between them is $30^\circ$, then $|\vec A \times \vec B|$ is
A. $6\sqrt{3}$
B. $3$
C. $6$ ✓ Correct
D. $12$
Solution: $|\vec A\times\vec B| = AB\sin\theta = 4\times 3\times\tfrac{1}{2} = 6$.
Q9 — Vector Multiplication · easy · numerical
If $|\vec A| = 4$, $|\vec B| = 3$ and the angle between them is $60^\circ$, then $\vec A \cdot \vec B$ is
A. $6\sqrt{3}$
B. $6$ ✓ Correct
C. $4$
D. $12$
Solution: $\vec A\cdot\vec B = AB\cos\theta = 4\times 3\times\tfrac{1}{2} = 6$.
Q10 — Vector Multiplication · easy · theory
If $\vec A \cdot \vec B = |\vec A \times \vec B|$, the angle between $\vec A$ and $\vec B$ is
A. $30^\circ$
B. $60^\circ$
C. $45^\circ$ ✓ Correct
D. $90^\circ$
Solution: $AB\cos\theta = AB\sin\theta \Rightarrow \tan\theta = 1 \Rightarrow \theta = 45^\circ$.
Q11 — Vector Multiplication · medium · theory
If $\vec A = \hat{i} + \hat{j}$ and $\vec B = \hat{j} + \hat{k}$, then $\vec A \times \vec B$ is
A. $\hat{i} + \hat{j} + \hat{k}$
B. $-\hat{i} + \hat{j} - \hat{k}$
C. $\hat{i} + \hat{j} - \hat{k}$
D. $\hat{i} - \hat{j} + \hat{k}$ ✓ Correct
Solution: Using the determinant rule with $A=(1,1,0)$, $B=(0,1,1)$: $\vec A\times\vec B = (1\cdot1-0\cdot1)\hat{i} - (1\cdot1-0\cdot0)\hat{j} + (1\cdot1-1\cdot0)\hat{k} = \hat{i}-\hat{j}+\hat{k}$.
Q12 — Vector Multiplication · easy · numerical
If $|\vec A| = 5$ and $|\vec B| = 3$, then $(\vec A + \vec B)\cdot(\vec A - \vec B)$ equals
A. $-16$
B. $16$ ✓ Correct
C. $8$
D. $34$
Solution: $(\vec A+\vec B)\cdot(\vec A-\vec B) = A^2 - B^2 = 25 - 9 = 16$.
Q13 — Vector Multiplication · medium · theory
A force $\vec F = 2\hat{j}$ N acts at the point $\vec r = \hat{i} + \hat{j}$ m. The torque about the origin is
A. Zero
B. $2\hat{i}$ N·m
C. $-2\hat{k}$ N·m
D. $2\hat{k}$ N·m ✓ Correct
Solution: $\vec\tau = \vec r\times\vec F = (\hat{i}+\hat{j})\times 2\hat{j} = 2(\hat{i}\times\hat{j}) = 2\hat{k}$ N·m.
Q14 — Vector Multiplication · medium · theory
For any vector $\vec A$, the value of $\vec A \times \vec A$ is
A. The null vector ✓ Correct
B. $\vec A$
C. $A^2$
D. A unit vector
Solution: The angle between a vector and itself is zero, and $\sin 0 = 0$, so $\vec A\times\vec A = \vec 0$.
Q15 — Vector Multiplication · easy · theory
$(\vec A \times \vec B)^2 + (\vec A \cdot \vec B)^2$ equals
A. $2A^2B^2$
B. $A^2 + B^2$
C. $(A+B)^2$
D. $A^2B^2$ ✓ Correct
Solution: $A^2B^2\sin^2\theta + A^2B^2\cos^2\theta = A^2B^2(\sin^2\theta + \cos^2\theta) = A^2B^2$.
Q16 — Vector Multiplication · easy · theory
A unit vector perpendicular to both $\hat{i}$ and $\hat{j}$ is
A. $\hat{k}$ ✓ Correct
B. $\hat{i}$
C. $\hat{i}+\hat{j}+\hat{k}$
D. $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$
Solution: $\hat{i}\times\hat{j} = \hat{k}$, which is perpendicular to both ($-\hat{k}$ also works).
Q17 — Vector Multiplication · medium · theory
The angle between $\vec A = 2\hat{i} + 2\hat{j}$ and the y-axis is
A. $30^\circ$
B. $45^\circ$ ✓ Correct
C. $60^\circ$
D. $90^\circ$
Solution: $\cos\theta = \frac{\vec A\cdot\hat{j}}{|\vec A|} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} \Rightarrow \theta = 45^\circ$.
Q18 — Vector Multiplication · medium · theory
Two non-zero vectors satisfy $\vec A \times \vec B = \vec 0$. The vectors are
A. Parallel or antiparallel ✓ Correct
B. At $45^\circ$
C. Perpendicular
D. Equal
Solution: $AB\sin\theta = 0$ with $A, B \neq 0$ forces $\sin\theta = 0$, i.e. $\theta = 0^\circ$ or $180^\circ$.
Q19 — Vector Multiplication · medium · numerical
If $\vec A = 2\hat{i} + 3\hat{j} + \hat{k}$ is parallel to $\vec B = 4\hat{i} + 6\hat{j} + \lambda\hat{k}$, then $\lambda$ is
A. $3$
B. $1$
C. $4$
D. $2$ ✓ Correct
Solution: Parallel vectors have proportional components: $\frac{4}{2} = \frac{6}{3} = \frac{\lambda}{1} = 2 \Rightarrow \lambda = 2$.
Q20 — Vector Multiplication · easy · numerical
A force $\vec F = (2\hat{i} + 3\hat{j})$ N acts on a body moving with velocity $\vec v = (\hat{i} + 2\hat{j})$ m/s. The power delivered is
A. $5$ W
B. $7$ W
C. $8$ W ✓ Correct
D. $4$ W
Solution: $P = \vec F\cdot\vec v = (2)(1) + (3)(2) = 8$ W.
Q21 — Vector Multiplication · easy · theory
If $\vec A \cdot \vec B = -|\vec A||\vec B|$, the angle between the vectors is
A. $45^\circ$
B. $0^\circ$
C. $180^\circ$ ✓ Correct
D. $90^\circ$
Solution: $\cos\theta = -1 \Rightarrow \theta = 180^\circ$ (antiparallel vectors).
Q22 — Vector Multiplication · medium · theory
The magnitude of $\vec A \times \vec B$ is maximum when the angle between $\vec A$ and $\vec B$ is
A. $45^\circ$
B. $90^\circ$ ✓ Correct
C. $0^\circ$
D. $180^\circ$
Solution: $|\vec A\times\vec B| = AB\sin\theta$ is maximum when $\sin\theta = 1$, i.e. $\theta = 90^\circ$.
Q23 — Vector Multiplication · medium · theory
A particle moves in the xy-plane. Its angular momentum $\vec L = \vec r \times \vec p$ about the origin points
A. Along the z-axis ✓ Correct
B. In the xy-plane
C. Along the x-axis
D. Along $\vec r$
Solution: The cross product of two vectors in the xy-plane is perpendicular to that plane, i.e. along $\pm\hat{k}$.
Q24 — Vector Multiplication · medium · theory
For any two vectors, $\vec A \cdot (\vec A \times \vec B)$ equals
A. $\vec B\cdot\vec A$
B. $A^2B\sin\theta$
C. $A^2B$
D. Zero ✓ Correct
Solution: $\vec A\times\vec B$ is perpendicular to $\vec A$, and the dot product of perpendicular vectors is zero.
Q25 — Vector Multiplication · medium · theory
The angle between $\vec A = \hat{i} - \hat{j}$ and $\vec B = \hat{j} - \hat{k}$ is
A. $60^\circ$
B. $45^\circ$
C. $120^\circ$ ✓ Correct
D. $90^\circ$
Solution: $\vec A\cdot\vec B = (1)(0) + (-1)(1) + (0)(-1) = -1$; $|\vec A| = |\vec B| = \sqrt{2}$. So $\cos\theta = \frac{-1}{2} \Rightarrow \theta = 120^\circ$.
Q26 — Vector Multiplication · medium · theory
If $|\vec A \times \vec B| = \sqrt{3}\,(\vec A \cdot \vec B)$, the angle between $\vec A$ and $\vec B$ is
A. $30^\circ$
B. $90^\circ$
C. $60^\circ$ ✓ Correct
D. $45^\circ$
Solution: $AB\sin\theta = \sqrt{3}AB\cos\theta \Rightarrow \tan\theta = \sqrt{3} \Rightarrow \theta = 60^\circ$. (A frequently asked JEE pattern.)
Q27 — Vector Multiplication · medium · theory
A particle in uniform circular motion experiences a centripetal force. The work done by this force in any time interval is
A. Negative
B. Depends on the speed
C. Zero ✓ Correct
D. Positive
Solution: The force is always perpendicular to the velocity, so $\vec F\cdot\vec v = 0$ at every instant — no work is done.
Q28 — Vector Multiplication · medium · theory
If $\vec A \cdot \vec B = 0$ and $\vec A \times \vec B = \vec 0$ with $\vec A \neq \vec 0$, then
A. $\vec B \perp \vec A$
B. $\vec B = \vec 0$ ✓ Correct
C. $|\vec B| = |\vec A|$
D. $\vec B \parallel \vec A$
Solution: The dot being zero needs $\theta = 90^\circ$ and the cross being zero needs $\theta = 0^\circ$ or $180^\circ$ — impossible together unless $\vec B$ itself is the null vector.
Q29 — Vector Multiplication · easy · theory
If $\hat{a}$ and $\hat{b}$ are unit vectors with $\hat{a}\cdot\hat{b} = 1$, then
A. They are opposite
B. They are perpendicular
C. They point in the same direction ✓ Correct
D. Nothing can be said
Solution: $\hat{a}\cdot\hat{b} = \cos\theta = 1 \Rightarrow \theta = 0^\circ$: identical directions.
Q30 — Vector Multiplication · medium · numerical
For A = 2î + 3ĵ and B = î + 4ĵ, the dot product A·B is:
A. 14 ✓ Correct
B. 2
C. 10
D. 5
Solution: A·B = 2(1) + 3(4) = 2 + 12 = 14.