Three Dimensional Geometry: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. Direction cosines of the line with ratios 1, –2, 2.
2. Direction cosines of the $z$-axis.
3. A line makes 60° and 45° with OX and OY: find $\gamma$.
4. Direction ratios of $\frac{x-2}{3}=\frac{2y+1}{4}=\frac{z}{-1}$.
5. Vector equation through (1, –1, 2) parallel to $2\hat{i}+\hat{j}-\hat{k}$.
6. Cartesian equation through (0, 0, 0) and (1, 2, 3).
7. Angle between lines with ratios (1, 1, 0) and (0, 1, 1).
8. Are ratios (2, 3, –4) and (4, 0, 2) perpendicular?
9. Direction cosines for ratios 3, 4, 0.
10. $k$ if ratios $(k,2,1)$ and $(1,-1,3)$ are perpendicular.

D. Short answer I (2 marks each)

1. Direction cosines of the line joining (1, 2, 3) and (3, 4, 4).
2. Angle between lines with direction ratios (1, 2, 2) and (2, 2, 1).
3. Show (1, 2, 3), (2, 3, 4), (4, 5, 6) are collinear.
4. Cartesian equation through (1, 2, –1) parallel to $\frac{x-3}{2}=\frac{y+1}{-1}=\frac{z}{4}$.
5. Cartesian form of $\vec{r}=(2\hat{i}-\hat{j})+\lambda (\hat{i}+3\hat{j}-\hat{k})$.
6. $p$ if $\frac{x-1}{2}=\frac{y-2}{p}=\frac{z}{3}$ and $\frac{x}{1}=\frac{y}{2}=\frac{z}{-2}$ are perpendicular.
7. Does (3, 5, 7) lie on $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$?
8. SD between $\vec{r}=\hat{i}+\hat{j}+\lambda (2\hat{i}-\hat{j}+\hat{k})$ and $\vec{r}=2\hat{i}+\hat{j}-\hat{k}+\mu (2\hat{i}-\hat{j}+\hat{k})$.

E. Short answer II (3 marks each)

1. SD between $\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$ and $\vec{r}=(2\hat{i}+4\hat{j}+5\hat{k})+\mu (2\hat{i}+3\hat{j}+4\hat{k})$.
2. Foot of the perpendicular from (1, 2, 3) to $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$, and the distance.
3. Image of (1, 6, 3) in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$.
4. Show that $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ intersect; find the point.
5. Vector equation of the line through (2, –1, 3) perpendicular to lines with ratios (1, 2, 3) and (3, 2, 1).
6. Distance of (2, 3, 4) from the line $\vec{r}=\lambda (\hat{i}+\hat{j}+\hat{k})$.

F. Long answer (5 marks each)

1. Find the shortest distance between $\vec{r}=(4\hat{i}-\hat{j})+\lambda (\hat{i}+2\hat{j}-3\hat{k})$ and $\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu (2\hat{i}+4\hat{j}-5\hat{k})$.
2. Find the foot of the perpendicular from P(2, –1, 5) to $\frac{x-11}{10}=\frac{y+2}{-4}=\frac{z+8}{-11}$, the length of the perpendicular and the image of P.

G. HOTS (higher-order thinking)

1. Why are direction ratios not unique, while direction cosines are (up to sign)?
2. Why is the SD between intersecting lines zero?
3. Can $\frac{1}{2},\frac{1}{2},\frac{1}{2}$ be direction cosines?
4. How can you test whether two lines in space are coplanar?

20 MCQs

1. Direction cosines of the $x$-axis
   (a) 0, 1, 0 (b) 0, 0, 1 (c) 1, 0, 0 (d) 1, 1, 1
2. For direction cosines, $l^2+m^2+n^2=$
   (a) 1 (b) 0 (c) 2 (d) 3
3. Direction cosines for ratios 2, 3, 6
   (a) 2, 3, 6 (b) $\frac{1}{2},\frac{1}{3},\frac{1}{6}$ (c) $\frac{2}{11},\frac{3}{11},\frac{6}{11}$ (d) $\frac{2}{7},\frac{3}{7},\frac{6}{7}$
4. A line equally inclined to the axes has direction cosines
   (a) 1, 1, 1 (b) $\pm \frac{1}{\sqrt{3}}$ each (c) $\frac{1}{3}$ each (d) $\pm \frac{1}{2}$ each
5. Direction ratios of $\frac{x-1}{2}=\frac{y+3}{-1}=\frac{z-4}{5}$
   (a) 1, –3, 4 (b) 2, –1, 5 (c) –1, 3, –4 (d) 2, 1, 5
6. A point on $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$
   (a) (2, 3, 4) (b) (–1, –2, –3) (c) (0, 0, 0) (d) (1, 2, 3)
7. Line through the origin along $\hat{i}+\hat{j}$
   (a) $\vec{r}=\lambda (\hat{i}+\hat{j})$ (b) $\vec{r}=\hat{i}+\hat{j}$ (c) $\vec{r}=\hat{i}+\hat{j}+\hat{k}$ (d) $\vec{r}\cdot (\hat{i}+\hat{j})=0$
8. Lines with ratios (1, 2, 3) and (3, 0, –1) are
   (a) parallel (b) coincident (c) perpendicular (d) skew
9. Angle between ratios (1, 0, 0) and (0, 1, 0)
   (a) 0 (b) $\pi$ (c) $\frac{\pi }{4}$ (d) $\frac{\pi }{2}$
10. SD between intersecting lines
    (a) 1 (b) 0 (c) infinite (d) –1
11. $\cos \alpha =0$, $\cos \beta =0$: the line is parallel to
    (a) the $x$-axis (b) the $y$-axis (c) the $z$-axis (d) the XY-plane
12. ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =$
    (a) 2 (b) 1 (c) 3 (d) 0
13. Cartesian form of $\vec{r}=(\hat{i}-\hat{j})+\lambda (2\hat{i}+\hat{k})$
    (a) $\frac{x-2}{1}=\frac{y}{-1}=\frac{z-1}{0}$ (b) $\frac{x-1}{2}=\frac{y+1}{0}=\frac{z}{1}$ (c) $\frac{x+1}{2}=\frac{y-1}{0}=\frac{z}{1}$ (d) $\frac{x}{2}=y=z$
14. $k$ for $\frac{x-1}{k}=\frac{y}{2}=\frac{z}{3}\perp \frac{x}{1}=\frac{y}{1}=\frac{z}{-1}$
    (a) –1 (b) 5 (c) 1 (d) 0
15. $\vec{r}={\vec{a}}_1+\lambda \vec{b}$ and $\vec{r}={\vec{a}}_2+\mu \vec{b}$ are
    (a) perpendicular (b) always intersecting (c) skew (d) parallel
16. Direction cosines of the line joining (1, 2, 3) and (4, 6, 3)
    (a) $\frac{3}{5},\frac{4}{5},0$ (b) 3, 4, 0 (c) $\frac{1}{5},\frac{2}{5},0$ (d) $\frac{4}{5},\frac{3}{5},0$
17. Angle between $\frac{x}{1}=\frac{y}{1}=\frac{z}{0}$ and $\frac{x}{1}=\frac{y}{-1}=\frac{z}{0}$
    (a) $\frac{\pi }{2}$ (b) $\frac{\pi }{4}$ (c) $\frac{\pi }{3}$ (d) 0
18. Direction cosines for ratios 1, –1, 0
    (a) 1, –1, 0 (b) 0, 1, –1 (c) $\frac{1}{2},-\frac{1}{2},0$ (d) $\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0$
19. Skew lines are
    (a) parallel (b) neither parallel nor intersecting (c) intersecting (d) coplanar
20. A line makes 90°, 60°, 30° with the axes; its direction cosines
    (a) $0,\frac{\sqrt{3}}{2},\frac{1}{2}$ (b) $1,\frac{1}{2},\frac{\sqrt{3}}{2}$ (c) $0,\frac{1}{2},\frac{\sqrt{3}}{2}$ (d) $0,\frac{1}{2},\frac{1}{2}$

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: A line equally inclined to the axes has direction cosines $\pm \frac{1}{\sqrt{3}}$ each. R: $l^2+m^2+n^2=1$.
2. A: Lines with ratios 1, 2, 3 and 3, 0, –1 are perpendicular. R: Perpendicular lines satisfy $a_1{a}_2+b_1{b}_2+c_1{c}_2=0$.
3. A: $\vec{r}=\hat{i}+\lambda (2\hat{i}+\hat{j})$ and $\vec{r}=\hat{j}+\mu (4\hat{i}+2\hat{j})$ are parallel. R: Lines with proportional direction ratios are parallel.
4. A: The SD between two distinct parallel lines is 0. R: Parallel lines do not intersect.
5. A: 1, 1, 1 are the direction cosines of a line. R: Direction cosines satisfy $l^2+m^2+n^2=1$.
6. A: $\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{3}$ passes through (2, 3, 4). R: Its direction ratios are 1, 2, 3.

Key: 1 (a) · 2 (a) · 3 (a) · 4 (d) · 5 (d) · 6 (b)

Answers to the practice set