Scalar (Dot) Product — MH-CET Physics MCQs with Solutions
Free MH-CET Physics Scalar (Dot) Product MCQs with step-by-step solutions (21 questions). Part of Mathematical Methods (Std 11). Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Scalar (Dot) Product · easy · theory
The scalar product of two vectors is defined as:
A. $AB\cos\theta$, which is a scalar ✓ Correct
B. $\dfrac{A}{B}\cos\theta$
C. $AB\tan\theta$
D. $AB\sin\theta$, which is a vector
Solution: It projects one vector onto the other and multiplies by the magnitude.
Q2 — Scalar (Dot) Product · easy · theory
The result of a scalar product of two vectors is:
A. A vector perpendicular to both
B. A scalar ✓ Correct
C. Always zero
D. A vector in the plane of the two
Solution: This is why it is also called the dot product.
Q3 — Scalar (Dot) Product · easy · theory
The scalar product is commutative, meaning that:
A. $\vec{A}\cdot\vec{B} = \vec{B}\cdot\vec{A}$ ✓ Correct
B. $\vec{A}\cdot\vec{B} = 0$
C. $\vec{A}\cdot\vec{B} = -\vec{B}\cdot\vec{A}$
D. $\vec{A}\cdot\vec{B} = \vec{A} \times \vec{B}$
Solution: The cosine of the angle is the same whichever vector is taken first.
Q4 — Scalar (Dot) Product · medium · theory
For the unit vectors along the coordinate axes:
A. $\hat{i}\cdot\hat{i} = 1$ and $\hat{i}\cdot\hat{j} = 0$ ✓ Correct
B. $\hat{i}\cdot\hat{i} = \hat{i}\cdot\hat{j} = 0$
C. $\hat{i}\cdot\hat{i} = 0$ and $\hat{i}\cdot\hat{j} = 1$
D. $\hat{i}\cdot\hat{i} = \hat{i}\cdot\hat{j} = 1$
Solution: Parallel unit vectors give one; perpendicular unit vectors give zero.
Q5 — Scalar (Dot) Product · easy · theory
Work done by a force is an example of a:
A. Scalar product of force and velocity
B. Vector sum
C. Scalar product of force and displacement ✓ Correct
D. Vector product of force and displacement
Solution: $W = \vec{F}\cdot\vec{s} = Fs\cos\theta$.
Q6 — Scalar (Dot) Product · easy · theory
Two non-zero vectors are perpendicular if their scalar product is:
A. Equal to $-AB$
B. Maximum
C. Zero ✓ Correct
D. Equal to $AB$
Solution: With $\theta = 90^\circ$, $\cos\theta = 0$.
Q7 — Scalar (Dot) Product · medium · theory
The scalar product $\vec{A}\cdot\vec{A}$ equals:
A. Zero
B. $2A$
C. $A^2$ ✓ Correct
D. $A$
Solution: A vector makes zero angle with itself, so the cosine is unity.
Q8 — Scalar (Dot) Product · hard · theory
The projection of $\vec{A}$ on $\vec{B}$ is given by:
A. $\vec{A}\cdot\vec{B}$
B. $|\vec{A}||\vec{B}|$
C. $\dfrac{\vec{A}\cdot\vec{B}}{|\vec{B}|}$ ✓ Correct
D. $\dfrac{\vec{A}\cdot\vec{B}}{|\vec{A}|}$
Solution: Dividing the dot product by the magnitude of $\vec{B}$ leaves $A\cos\theta$.
Q9 — Scalar (Dot) Product · medium · numerical
The vectors $2\hat{i} + 3\hat{j}$ and $3\hat{i} - 2\hat{j}$ are:
A. Perpendicular to each other ✓ Correct
B. Inclined at $45^\circ$
C. Parallel to each other
D. Antiparallel
Solution: $\vec{A}\cdot\vec{B} = (2)(3) + (3)(-2) = 0$, so the angle is $90^\circ$.
Q10 — Scalar (Dot) Product · hard · numerical
The angle between the vectors $\hat{i} + \hat{j}$ and $\hat{j} + \hat{k}$ is:
A. $30^\circ$
B. $45^\circ$
C. $90^\circ$
D. $60^\circ$ ✓ Correct
Solution: $\cos\theta = \dfrac{(1)(0) + (1)(1) + (0)(1)}{\sqrt{2}\sqrt{2}} = \dfrac{1}{2}$, so $\theta = 60^\circ$.
Q11 — Scalar (Dot) Product · medium · numerical
The scalar product of $3\hat{i} + 4\hat{j}$ and $4\hat{i} + 3\hat{j}$ is:
A. $24$ ✓ Correct
B. $7$
C. $12$
D. $25$
Solution: $(3)(4) + (4)(3) = 12 + 12 = 24$.
Q12 — Scalar (Dot) Product · hard · numerical
The scalar projection of $3\hat{i} + 4\hat{j}$ onto $\hat{i} + \hat{j}$ is:
A. $\dfrac{5}{\sqrt{2}}$
B. $\dfrac{7}{2}$
C. $\dfrac{\sqrt{2}}{7}$
D. $\dfrac{7}{\sqrt{2}}$ ✓ Correct
Solution: Projection $= \dfrac{\vec{A}\cdot\vec{B}}{|\vec{B}|} = \dfrac{3 + 4}{\sqrt{2}} = \dfrac{7}{\sqrt{2}}$.
Q13 — Scalar (Dot) Product · easy · numerical
The scalar product of $2\hat{i}$ and $3\hat{j}$ is:
A. $6\hat{k}$
B. Zero ✓ Correct
C. $6$
D. $5$
Solution: The two vectors are perpendicular, so their dot product vanishes.
Q14 — Scalar (Dot) Product · medium · numerical
A force $\vec{F} = 5\hat{i} + 3\hat{j}\text{ N}$ acts through a displacement $\vec{s} = 2\hat{i}\text{ m}$. The work done is:
A. $8\text{ J}$
B. $16\text{ J}$
C. $6\text{ J}$
D. $10\text{ J}$ ✓ Correct
Solution: $W = \vec{F}\cdot\vec{s} = (5)(2) + (3)(0) = 10\text{ J}$.
Q15 — Scalar (Dot) Product · medium · numerical
For $\vec{A} = 3\hat{i} + 4\hat{j}$, the value of $\vec{A}\cdot\vec{A}$ is:
A. $5$
B. $7$
C. $25$ ✓ Correct
D. $12$
Solution: $\vec{A}\cdot\vec{A} = A^2 = 5^2 = 25$.
Q16 — Scalar (Dot) Product · hard · numerical
Two vectors of magnitudes $3$ and $8$ have a scalar product of $12$. The angle between them is:
A. $60^\circ$ ✓ Correct
B. $90^\circ$
C. $45^\circ$
D. $30^\circ$
Solution: $\cos\theta = \dfrac{12}{3 \times 8} = 0.5$, so $\theta = 60^\circ$.
Q17 — Scalar (Dot) Product · easy · numerical
The value of $\hat{i}\cdot\hat{i}$ is:
A. $1$ ✓ Correct
B. $-1$
C. $0$
D. $\hat{i}$
Solution: A unit vector dotted with itself gives the square of its magnitude, which is one.
Q18 — Scalar (Dot) Product · easy · numerical
The value of $\hat{i}\cdot\hat{j}$ is:
A. $-1$
B. $1$
C. $0$ ✓ Correct
D. $\hat{k}$
Solution: The two unit vectors are perpendicular.
Q19 — Scalar (Dot) Product · medium · numerical
The scalar product of $2\hat{i} + 3\hat{j} + 4\hat{k}$ and $\hat{i} - \hat{j} + \hat{k}$ is:
A. $3$ ✓ Correct
B. $5$
C. $9$
D. $-3$
Solution: $(2)(1) + (3)(-1) + (4)(1) = 2 - 3 + 4 = 3$.
Q20 — Scalar (Dot) Product · medium · numerical
If $\vec{A}\cdot\vec{B} = AB$, the angle between the vectors is:
A. $90^\circ$
B. $45^\circ$
C. $0^\circ$ ✓ Correct
D. $180^\circ$
Solution: The dot product reaches its maximum value $AB$ only when $\cos\theta = 1$.
Q21 — Scalar (Dot) Product · medium · numerical
A force of $10\text{ N}$ moves a body through $5\text{ m}$ at $60^\circ$ to the force. The work done is:
A. $50\text{ J}$
B. $25\text{ J}$ ✓ Correct
C. $43.3\text{ J}$
D. $12.5\text{ J}$
Solution: $W = Fs\cos\theta = 10 \times 5 \times 0.5 = 25\text{ J}$.