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Vector Multiplication — JEE Main Physics MCQs with Solutions
Free JEE Main Physics Vector Multiplication MCQs with step-by-step solutions (28 questions). Part of Vectors. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Vector Multiplication · medium · theory
The angle between $\vec A = \hat{i} + \hat{j}$ and $\vec B = \hat{i}$ is
A. $45^\circ$ ✓ Correct
B. $30^\circ$
C. $90^\circ$
D. $60^\circ$
Solution: $\cos\theta = \frac{\vec A\cdot\vec B}{AB} = \frac{1}{\sqrt{2}\cdot 1} \Rightarrow \theta = 45^\circ$.
Q2 — 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. $3$ ✓ Correct
C. $6$
D. $\frac{2}{3}$
Solution: Perpendicular $\Rightarrow \vec A\cdot\vec B = 0$: $6 - 2\lambda = 0 \Rightarrow \lambda = 3$.
Q3 — 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. $3$
B. $2$ ✓ Correct
C. $6$
D. $\sqrt{6}$
Solution: Projection $= \frac{\vec A\cdot\vec B}{|\vec B|} = \frac{2+2+2}{\sqrt{4+1+4}} = \frac{6}{3} = 2$.
Q4 — Vector Multiplication · medium · theory
$\hat{i}\times\hat{j}$ equals
A. $0$
B. $\hat{k}$ ✓ Correct
C. $-\hat{k}$
D. $\hat{j}$
Solution: By the right-hand rule for the cyclic order $x\to y\to z$: $\hat{i}\times\hat{j} = \hat{k}$.
Q5 — 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 $A=(1,1,0)$, $B=(0,1,1)$: $\vec A\times\vec B = (1\cdot1-0\cdot1)\hat{i} - (1\cdot1-0\cdot0)\hat{j} + (1\cdot1-1\cdot0)\hat{k} = \hat{i}-\hat{j}+\hat{k}$.
Q6 — 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. $2\hat{k}$ N·m ✓ Correct
C. $-2\hat{k}$ N·m
D. $2\hat{i}$ N·m
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.
Q7 — Vector Multiplication · medium · theory
For any vector $\vec A$, the value of $\vec A \times \vec A$ is
A. $\vec A$
B. $A^2$
C. A unit vector
D. The null vector ✓ Correct
Solution: The angle between a vector and itself is zero, and $\sin 0 = 0$, so $\vec A\times\vec A = \vec 0$.
Q8 — Vector Multiplication · medium · theory
The angle between $\vec A = 2\hat{i} + 2\hat{j}$ and the y-axis is
A. $90^\circ$
B. $60^\circ$
C. $45^\circ$ ✓ Correct
D. $30^\circ$
Solution: $\cos\theta = \frac{\vec A\cdot\hat{j}}{|\vec A|} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} \Rightarrow \theta = 45^\circ$.
Q9 — Vector Multiplication · medium · theory
Two non-zero vectors satisfy $\vec A \times \vec B = \vec 0$. The vectors are
A. At $45^\circ$
B. Parallel or antiparallel ✓ Correct
C. Perpendicular
D. Equal
Solution: $AB\sin\theta = 0$ with $A, B \neq 0$ forces $\sin\theta = 0$, i.e. $\theta = 0^\circ$ or $180^\circ$.
Q10 — 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. $4$
B. $3$
C. $1$
D. $2$ ✓ Correct
Solution: Parallel vectors have proportional components: $\frac{4}{2} = \frac{6}{3} = \frac{\lambda}{1} = 2 \Rightarrow \lambda = 2$.
Q11 — 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. $180^\circ$
B. $45^\circ$
C. $0^\circ$
D. $90^\circ$ ✓ Correct
Solution: $|\vec A\times\vec B| = AB\sin\theta$ is maximum when $\sin\theta = 1$, i.e. $\theta = 90^\circ$.
Q12 — 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 $\vec r$
B. In the xy-plane
C. Along the x-axis
D. Along the z-axis ✓ Correct
Solution: The cross product of two vectors in the xy-plane is perpendicular to that plane, i.e. along $\pm\hat{k}$.
Q13 — Vector Multiplication · medium · theory
For any two vectors, $\vec A \cdot (\vec A \times \vec B)$ equals
A. Zero ✓ Correct
B. $A^2B\sin\theta$
C. $A^2B$
D. $\vec B\cdot\vec A$
Solution: $\vec A\times\vec B$ is perpendicular to $\vec A$, and the dot product of perpendicular vectors is zero.
Q14 — Vector Multiplication · medium · theory
The angle between $\vec A = \hat{i} - \hat{j}$ and $\vec B = \hat{j} - \hat{k}$ is
A. $45^\circ$
B. $90^\circ$
C. $60^\circ$
D. $120^\circ$ ✓ Correct
Solution: $\vec A\cdot\vec B = (1)(0) + (-1)(1) + (0)(-1) = -1$; $|\vec A| = |\vec B| = \sqrt{2}$. So $\cos\theta = \frac{-1}{2} \Rightarrow \theta = 120^\circ$.
Q15 — 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. $45^\circ$
B. $90^\circ$
C. $30^\circ$
D. $60^\circ$ ✓ Correct
Solution: $AB\sin\theta = \sqrt{3}AB\cos\theta \Rightarrow \tan\theta = \sqrt{3} \Rightarrow \theta = 60^\circ$. (A frequently asked JEE pattern.)
Q16 — 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. Zero ✓ Correct
B. Depends on the speed
C. Positive
D. Negative
Solution: The force is always perpendicular to the velocity, so $\vec F\cdot\vec v = 0$ at every instant — no work is done.
Q17 — 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.
Q18 — Vector Multiplication · medium · numerical
For A = 2î + 3ĵ and B = î + 4ĵ, the dot product A·B is:
A. 14 ✓ Correct
B. 10
C. 5
D. 2
Solution: A·B = 2(1) + 3(4) = 2 + 12 = 14.
Q19 — Vector Multiplication · hard · numerical
Two vectors of magnitude 4 and 5 have a dot product of 10. The angle between them is:
A. 60° ✓ Correct
B. 90°
C. 45°
D. 30°
Solution: cosθ = 10/(4×5) = 0.5 ⇒ θ = 60°.
Q20 — Vector Multiplication · medium · numerical
The magnitude of the cross product of two vectors of magnitude 3 and 6 at 30° is:
A. 4.5
B. 18
C. 9 ✓ Correct
D. 15.6
Solution: |A×B| = AB sinθ = 3×6×0.5 = 9.
Q21 — Vector Multiplication · medium · numerical
If A·B = 0 for two non-zero vectors, the angle between them is:
A. 0°
B. 90° ✓ Correct
C. 45°
D. 180°
Solution: cosθ = 0 ⇒ θ = 90° (perpendicular).
Q22 — Vector Multiplication · medium · numerical
For A = î + 2ĵ + 3k̂ and B = 2î + ĵ + k̂, A·B is:
A. 9
B. 7 ✓ Correct
C. 5
D. 6
Solution: A·B = 1(2) + 2(1) + 3(1) = 2 + 2 + 3 = 7.
Q23 — Vector Multiplication · medium · numerical
Two vectors of magnitude 2 and 2 are parallel. Their cross-product magnitude is:
A. 8
B. 0 ✓ Correct
C. 4
D. 2
Solution: sin0° = 0 ⇒ |A×B| = 0 for parallel vectors.
Q24 — Vector Multiplication · hard · numerical
For vectors A and B, |A × B| = √3 (A·B). The angle between them is:
A. 45°
B. 90°
C. 30°
D. 60° ✓ Correct
Solution: ABsinθ = √3 ABcosθ ⇒ tanθ = √3 ⇒ θ = 60°.
Q25 — Vector Multiplication · hard · numerical
If A = 2î + 3ĵ + k̂ and B = î − ĵ + 2k̂, then A × B is:
A. 7î − 3ĵ − 5k̂ ✓ Correct
B. 7î + 3ĵ − 5k̂
C. 5î − 3ĵ − 7k̂
D. −7î + 3ĵ + 5k̂
Solution: Determinant: î(6−(−1)) − ĵ(4−1) + k̂(−2−3) = 7î − 3ĵ − 5k̂.
Q26 — Vector Multiplication · hard · numerical
The vectors A = 2î + aĵ + k̂ and B = 4î − 2ĵ − 2k̂ are perpendicular. Then a =
A. −3
B. 2
C. 1
D. 3 ✓ Correct
Solution: A·B = 8 − 2a − 2 = 0 ⇒ a = 3.
Q27 — Vector Multiplication · hard · numerical
A force F = (3î + 4ĵ) N acts at the point r = (2î + 3ĵ) m. The torque about the origin is:
A. +k̂ N·m
B. −17k̂ N·m
C. −k̂ N·m ✓ Correct
D. 17k̂ N·m
Solution: τ = r × F = (2×4 − 3×3)k̂ = (8 − 9)k̂ = −k̂ N·m.
Q28 — Vector Multiplication · hard · numerical
The component (projection) of A = 3î + 4ĵ along B = î + ĵ is:
A. √7
B. 7
C. 5
D. 7/√2 ✓ Correct
Solution: A·B̂ = (3 + 4)/√2 = 7/√2 ≈ 4.95.