Direction Cosine — NEET Physics MCQs with Solutions
Free NEET Physics Direction Cosine 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 — Direction Cosine · medium · numerical
If a line makes angles $\alpha$, $\beta$, $\gamma$ with the x, y and z axes, then $\cos^2\alpha + \cos^2\beta + \cos^2\gamma$ equals
A. $0$
B. $1$ ✓ Correct
C. $2$
D. $3$
Solution: The direction cosines $(l, m, n)$ are the components of a unit vector, so $l^2 + m^2 + n^2 = 1$.
Q2 — Direction Cosine · easy · numerical
For a line making angles $\alpha$, $\beta$, $\gamma$ with the coordinate axes, $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ equals
A. $0$
B. $3$
C. $1$
D. $2$ ✓ Correct
Solution: $\sum\sin^2 = \sum(1 - \cos^2) = 3 - 1 = 2$.
Q3 — Direction Cosine · medium · theory
A vector makes equal angles with the three coordinate axes. Each angle is
A. $60^\circ$
B. $\cos^{-1}\left(\frac{1}{3}\right)$
C. $45^\circ$
D. $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$ ✓ Correct
Solution: $3\cos^2\alpha = 1 \Rightarrow \cos\alpha = \frac{1}{\sqrt{3}} \Rightarrow \alpha \approx 54.7^\circ$. (Watch the trap: it is $\frac{1}{\sqrt{3}}$, not $\frac{1}{3}$.)
Q4 — Direction Cosine · medium · theory
The direction cosines of the vector $2\hat{i} + 3\hat{j} + 6\hat{k}$ are
A. $2, 3, 6$
B. $\frac{1}{7}, \frac{1}{7}, \frac{1}{7}$
C. $\frac{2}{7}, \frac{3}{7}, \frac{6}{7}$ ✓ Correct
D. $\frac{2}{11}, \frac{3}{11}, \frac{6}{11}$
Solution: Magnitude $= \sqrt{4+9+36} = 7$; direction cosines are the components divided by the magnitude.
Q5 — Direction Cosine · medium · theory
The direction cosines of the vector $\hat{i} + 2\hat{j} + 2\hat{k}$ are
A. $\frac{1}{9}, \frac{4}{9}, \frac{4}{9}$
B. $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$ ✓ Correct
C. $\frac{1}{5}, \frac{2}{5}, \frac{2}{5}$
D. $1, 2, 2$
Solution: Magnitude $= \sqrt{1+4+4} = 3$, so the direction cosines are $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$.
Q6 — Direction Cosine · easy · theory
Can a line make angles $30^\circ$, $45^\circ$ and $60^\circ$ with the x, y and z axes respectively?
A. Yes, but only in the first octant
B. Yes, always
C. No, because $\cos^2 30^\circ + \cos^2 45^\circ + \cos^2 60^\circ \neq 1$ ✓ Correct
D. Only if the line passes through the origin
Solution: $\frac{3}{4} + \frac{1}{2} + \frac{1}{4} = \frac{3}{2} \neq 1$ — such a line cannot exist.
Q7 — Direction Cosine · medium · theory
A line makes $60^\circ$ with the x-axis and $60^\circ$ with the y-axis. The angle it makes with the z-axis is
A. $45^\circ$ ✓ Correct
B. $30^\circ$
C. $60^\circ$
D. $90^\circ$
Solution: $\cos^2\gamma = 1 - \frac{1}{4} - \frac{1}{4} = \frac{1}{2} \Rightarrow \cos\gamma = \frac{1}{\sqrt{2}} \Rightarrow \gamma = 45^\circ$.
Q8 — Direction Cosine · medium · numerical
The direction cosines of the x-axis are
A. $(0, 1, 0)$
B. $(1, 1, 1)$
C. $(1, 0, 0)$ ✓ Correct
D. $(0, 0, 1)$
Solution: The x-axis makes $0^\circ$ with itself and $90^\circ$ with y and z: $(\cos 0, \cos 90^\circ, \cos 90^\circ) = (1, 0, 0)$.
Q9 — Direction Cosine · medium · theory
A line has direction ratios $1, -2, 2$. Its direction cosines are
A. $\pm\left(\frac{1}{9}, -\frac{4}{9}, \frac{4}{9}\right)$
B. $\pm\left(\frac{1}{3}, -\frac{2}{3}, \frac{2}{3}\right)$ ✓ Correct
C. $\pm\left(\frac{1}{5}, -\frac{2}{5}, \frac{2}{5}\right)$
D. $(1, -2, 2)$
Solution: Divide the ratios by $\sqrt{1+4+4} = 3$. The sign pair reflects the two opposite senses of the line.
Q10 — Direction Cosine · medium · theory
A vector $\vec r$ with direction cosines $l, m, n$ can be written as
A. $|\vec r|^2(l\hat{i} + m\hat{j} + n\hat{k})$
B. $\frac{l\hat{i} + m\hat{j} + n\hat{k}}{|\vec r|}$
C. $|\vec r|(l\hat{i} + m\hat{j} + n\hat{k})$ ✓ Correct
D. $l\hat{i} + m\hat{j} + n\hat{k}$ always
Solution: $(l, m, n)$ form the unit vector along $\vec r$; multiplying by the magnitude reconstructs the vector.
Q11 — Direction Cosine · medium · theory
The angle between two lines with direction cosines $(1, 0, 0)$ and $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0\right)$ is
A. $90^\circ$
B. $45^\circ$ ✓ Correct
C. $30^\circ$
D. $60^\circ$
Solution: $\cos\theta = l_1l_2 + m_1m_2 + n_1n_2 = \frac{1}{\sqrt{2}} \Rightarrow \theta = 45^\circ$.
Q12 — Direction Cosine · medium · theory
The direction cosines of the vector $\hat{i} + \hat{j}$ are
A. $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$
B. $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0\right)$ ✓ Correct
C. $(1, 1, 0)$
D. $\left(\frac{1}{2}, \frac{1}{2}, 0\right)$
Solution: Magnitude $\sqrt{2}$: cosines $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0\right)$ — $45^\circ$ with x and y, $90^\circ$ with z.
Q13 — Direction Cosine · medium · theory
Can $\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$ be the direction cosines of a line?
A. Only if the vector is a unit vector
B. No, because the squares add to $\frac{3}{4}$, not $1$ ✓ Correct
C. Yes
D. Only for lines in the first octant
Solution: $\frac{1}{4}+\frac{1}{4}+\frac{1}{4} = \frac{3}{4} \neq 1$, violating $l^2+m^2+n^2 = 1$.
Q14 — Direction Cosine · easy · theory
If a vector makes $\gamma = 90^\circ$ with the z-axis, the vector lies
A. In the xy-plane ✓ Correct
B. Along the z-axis
C. Along the x-axis
D. In the yz-plane
Solution: $n = \cos 90^\circ = 0$ means no z-component — the vector lies entirely in the xy-plane.
Q15 — Direction Cosine · medium · theory
A line makes $90^\circ$ with both the x-axis and the y-axis. The angle with the z-axis is
A. $60^\circ$
B. $90^\circ$
C. $45^\circ$
D. $0^\circ$ (or $180^\circ$) ✓ Correct
Solution: $\cos^2\gamma = 1 - 0 - 0 = 1 \Rightarrow \gamma = 0^\circ$ or $180^\circ$: the line is along the z-axis.
Q16 — Direction Cosine · medium · theory
The angle that the body diagonal of a cube makes with any of its edges is
A. $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$ ✓ Correct
B. $60^\circ$
C. $\cos^{-1}\left(\frac{1}{3}\right)$
D. $45^\circ$
Solution: The diagonal $a\hat{i}+a\hat{j}+a\hat{k}$ has direction cosine $\frac{1}{\sqrt{3}}$ with each axis.
Q17 — Direction Cosine · medium · theory
The angle between two body diagonals of a cube is
A. $60^\circ$
B. $90^\circ$
C. $\cos^{-1}\left(\frac{1}{3}\right)$ ✓ Correct
D. $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$
Solution: Take diagonals $(1,1,1)$ and $(1,1,-1)$: $\cos\theta = \frac{1+1-1}{3} = \frac{1}{3}$. (A famous JEE result $\approx 70.5^\circ$.)
Q18 — Direction Cosine · medium · theory
A unit vector has direction cosines $\left(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\right)$. The vector is
A. $\frac{2\hat{i} + 3\hat{j} + 6\hat{k}}{49}$
B. $7(2\hat{i} + 3\hat{j} + 6\hat{k})$
C. $\frac{2\hat{i} + 3\hat{j} + 6\hat{k}}{7}$ ✓ Correct
D. $2\hat{i} + 3\hat{j} + 6\hat{k}$
Solution: A unit vector IS its direction cosines: $\frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} + \frac{6}{7}\hat{k}$.
Q19 — Direction Cosine · medium · theory
The direction ratios of a line are $1 : 1 : \sqrt{2}$. The angle the line makes with the z-axis is
A. $60^\circ$
B. $30^\circ$
C. $45^\circ$ ✓ Correct
D. $90^\circ$
Solution: Magnitude $= \sqrt{1+1+2} = 2$, so $n = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}} \Rightarrow \gamma = 45^\circ$ (and $60^\circ$ with x and y).
Q20 — Direction Cosine · medium · theory
A vector makes $45^\circ$ with the x-axis and $60^\circ$ with the y-axis. The angle with the z-axis is
A. $30^\circ$
B. $45^\circ$
C. $90^\circ$
D. $60^\circ$ ✓ Correct
Solution: $\cos^2\gamma = 1 - \frac{1}{2} - \frac{1}{4} = \frac{1}{4} \Rightarrow \cos\gamma = \frac{1}{2} \Rightarrow \gamma = 60^\circ$.
Q21 — Direction Cosine · medium · theory
If a line makes equal angles $\alpha$ with all three axes, then $\cos\alpha$ equals
A. $\pm\frac{1}{3}$
B. $\pm\frac{1}{\sqrt{2}}$
C. $\pm\frac{1}{\sqrt{3}}$ ✓ Correct
D. $1$
Solution: $3\cos^2\alpha = 1 \Rightarrow \cos\alpha = \pm\frac{1}{\sqrt{3}}$. ($\frac{1}{3}$ is the classic wrong answer.)
Q22 — Direction Cosine · medium · numerical
The direction cosines of $-\hat{k}$ are
A. $(0, 0, 1)$
B. $(1, 1, 0)$
C. $(-1, -1, -1)$
D. $(0, 0, -1)$ ✓ Correct
Solution: The vector $-\hat{k}$ makes $90^\circ$ with x and y, and $180^\circ$ with z: $(\,0, 0, \cos 180^\circ) = (0,0,-1)$.
Q23 — Direction Cosine · medium · theory
Direction ratios differ from direction cosines in that direction ratios
A. Are the sines of the angles
B. Are always positive
C. Must always be unit values
D. Are any numbers proportional to the cosines — their squares need not add to 1 ✓ Correct
Solution: Ratios only fix the direction up to scale; dividing by their magnitude gives the actual cosines.
Q24 — Direction Cosine · medium · theory
A line is perpendicular to two lines whose direction ratios are $(1,0,0)$ and $(0,1,0)$. Its direction is along
A. The x-axis
B. The z-axis ✓ Correct
C. The xy-plane
D. The line $x = y$
Solution: Perpendicular to both x and y directions means along their cross product: $\hat{i}\times\hat{j} = \hat{k}$.
Q25 — Direction Cosine · medium · theory
The angle that the vector $\hat{i} + \hat{j} + \hat{k}$ makes with the y-axis is
A. $\cos^{-1}\left(\frac{1}{3}\right)$
B. $45^\circ$
C. $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$ ✓ Correct
D. $60^\circ$
Solution: $\cos\beta = \frac{\vec r\cdot\hat{j}}{|\vec r|} = \frac{1}{\sqrt{3}}$ — the same for all three axes by symmetry.
Q26 — Direction Cosine · medium · theory
If $(l, m, n)$ are the direction cosines of a line, then $(-l, -m, -n)$ represents
A. An impossible set
B. A perpendicular line
C. A different line entirely
D. The same line traversed in the opposite sense ✓ Correct
Solution: Negating all three cosines reverses the direction along the same line — both triplets satisfy $l^2+m^2+n^2=1$.
Q27 — Direction Cosine · medium · theory
The x-component of a vector $\vec A$ in terms of its magnitude and direction cosine $l$ is
A. $A_x = |\vec A|\,l$ ✓ Correct
B. $A_x = \frac{l}{|\vec A|}$
C. $A_x = |\vec A|\sin\alpha$
D. $A_x = l^2|\vec A|$
Solution: $l = \cos\alpha = \frac{A_x}{|\vec A|}$, so $A_x = |\vec A|\,l$ — components are magnitude × direction cosine.
Q28 — Direction Cosine · medium · theory
Two lines have direction cosines $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0\right)$ and $\left(\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}, 0\right)$. The angle between them is
A. $45^\circ$
B. $90^\circ$ ✓ Correct
C. $60^\circ$
D. $0^\circ$
Solution: $\cos\theta = \frac{1}{2} - \frac{1}{2} + 0 = 0 \Rightarrow \theta = 90^\circ$.
Q29 — Direction Cosine · medium · theory
A line makes $120^\circ$ with the x-axis and $60^\circ$ with the y-axis. The acute angle it makes with the z-axis is
A. $90^\circ$
B. $60^\circ$
C. $45^\circ$ ✓ Correct
D. $30^\circ$
Solution: $\cos^2\gamma = 1 - \frac{1}{4} - \frac{1}{4} = \frac{1}{2} \Rightarrow \gamma = 45^\circ$ (taking the acute value).
Q30 — Direction Cosine · medium · numerical
For the vector 2î + 3ĵ + 6k̂, the magnitude is:
A. 11
B. 49
C. 5
D. 7 ✓ Correct
Solution: √(4+9+36) = √49 = 7.