Direction Cosine — JEE Main Physics MCQs with Solutions
Free JEE Main Physics Direction Cosine MCQs with step-by-step solutions (35 questions). Part of Vectors. Practise online on Prepizo — no login needed.
▶ Practise Direction Cosine online (free)
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. $3$
C. $2$
D. $1$ ✓ Correct
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. $2$ ✓ Correct
D. $1$
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. $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$ ✓ Correct
B. $\cos^{-1}\left(\frac{1}{3}\right)$
C. $45^\circ$
D. $60^\circ$
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. $\frac{1}{7}, \frac{1}{7}, \frac{1}{7}$
B. $2, 3, 6$
C. $\frac{2}{11}, \frac{3}{11}, \frac{6}{11}$
D. $\frac{2}{7}, \frac{3}{7}, \frac{6}{7}$ ✓ Correct
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. $\frac{1}{5}, \frac{2}{5}, \frac{2}{5}$
C. $1, 2, 2$
D. $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$ ✓ Correct
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. Only if the line passes through the origin
B. Yes, always
C. No, because $\cos^2 30^\circ + \cos^2 45^\circ + \cos^2 60^\circ \neq 1$ ✓ Correct
D. Yes, but only in the first octant
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. $30^\circ$
B. $60^\circ$
C. $90^\circ$
D. $45^\circ$ ✓ Correct
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. $(1, 0, 0)$ ✓ Correct
C. $(1, 1, 1)$
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. $(1, -2, 2)$
C. $\pm\left(\frac{1}{5}, -\frac{2}{5}, \frac{2}{5}\right)$
D. $\pm\left(\frac{1}{3}, -\frac{2}{3}, \frac{2}{3}\right)$ ✓ Correct
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. $|\vec r|(l\hat{i} + m\hat{j} + n\hat{k})$ ✓ Correct
C. $\frac{l\hat{i} + m\hat{j} + n\hat{k}}{|\vec r|}$
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. $60^\circ$
B. $45^\circ$ ✓ Correct
C. $90^\circ$
D. $30^\circ$
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. $\left(\frac{1}{2}, \frac{1}{2}, 0\right)$
C. $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0\right)$ ✓ Correct
D. $(1, 1, 0)$
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. Yes
B. Only if the vector is a unit vector
C. Only for lines in the first octant
D. No, because the squares add to $\frac{3}{4}$, not $1$ ✓ Correct
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. Along the z-axis
B. Along the x-axis
C. In the yz-plane
D. In the xy-plane ✓ Correct
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. $45^\circ$
B. $90^\circ$
C. $0^\circ$ (or $180^\circ$) ✓ Correct
D. $60^\circ$
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. $45^\circ$
B. $60^\circ$
C. $\cos^{-1}\left(\frac{1}{3}\right)$
D. $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$ ✓ Correct
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. $90^\circ$
B. $60^\circ$
C. $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$
D. $\cos^{-1}\left(\frac{1}{3}\right)$ ✓ Correct
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. $\frac{2\hat{i} + 3\hat{j} + 6\hat{k}}{7}$ ✓ Correct
B. $2\hat{i} + 3\hat{j} + 6\hat{k}$
C. $7(2\hat{i} + 3\hat{j} + 6\hat{k})$
D. $\frac{2\hat{i} + 3\hat{j} + 6\hat{k}}{49}$
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. $30^\circ$
B. $45^\circ$ ✓ Correct
C. $60^\circ$
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. $90^\circ$
B. $60^\circ$ ✓ Correct
C. $45^\circ$
D. $30^\circ$
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. $1$
B. $\pm\frac{1}{3}$
C. $\pm\frac{1}{\sqrt{2}}$
D. $\pm\frac{1}{\sqrt{3}}$ ✓ Correct
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. $(-1, -1, -1)$
B. $(1, 1, 0)$
C. $(0, 0, 1)$
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 any numbers proportional to the cosines — their squares need not add to 1 ✓ Correct
B. Are always positive
C. Are the sines of the angles
D. Must always be unit values
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 xy-plane
C. The line $x = y$
D. The z-axis ✓ Correct
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. $60^\circ$
B. $\cos^{-1}\left(\frac{1}{3}\right)$
C. $45^\circ$
D. $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$ ✓ Correct
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. A different line entirely
B. A perpendicular line
C. The same line traversed in the opposite sense ✓ Correct
D. An impossible set
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. $A_x = |\vec A|\sin\alpha$
B. $A_x = |\vec A|\,l$ ✓ Correct
C. $A_x = l^2|\vec A|$
D. $A_x = \frac{l}{|\vec A|}$
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. $0^\circ$
B. $90^\circ$ ✓ Correct
C. $45^\circ$
D. $60^\circ$
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. $30^\circ$
D. $45^\circ$ ✓ Correct
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. 49
B. 7 ✓ Correct
C. 5
D. 11
Solution: √(4+9+36) = √49 = 7.