Sample QuestionsThree Dimensional Geometry questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The direction cosines of the line passing through the two points $(-2,4,-5)$ and $(1,2,3)$ is :
- A
$\frac{3}{\sqrt{70}}, \frac{2}{\sqrt{70}}, \frac{8}{\sqrt{70}}$
- ✓
$\frac{3}{\sqrt{77}}, \frac{-2}{\sqrt{77}}, \frac{8}{\sqrt{77}}$
- C
$\frac{2}{\sqrt{77}}, \frac{-3}{\sqrt{77}}, \frac{8}{\sqrt{77}}$
- D
$\frac{8}{\sqrt{13}}, \frac{-2}{\sqrt{13}}, \frac{3}{\sqrt{13}}$.
Answer: B.
View full solution →The direction cosine of $y$-axis is:
- A
$0,0,0$
- B
$1,0,0$
- ✓
$0,1,0$
- D
$0,0,1$.
Answer: C.
View full solution →The distance of the point $(\alpha, \beta, \gamma)$ from $y$-axis is :
Answer: D.
View full solution →If a line makes angles of $90^{\circ}, 135^{\circ}, 45^{\circ}$ with $x, y$ and $z$ axes, then its direction cosines will be :
- A
$\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}, 0$
- ✓
$0,-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$
- C
$\frac{1}{2},-\frac{1}{2}, 0$
- D
$0, \frac{1}{2},-\frac{1}{2}$
Answer: B.
View full solution →If the lengths of projections of a line segment on the axes be respectively $3,4,12$, then the length of the line segment is:
Answer: D.
View full solution →Write the condition of three lines with direction cosines $l_1, m_1, n_1 ; l_2, m_2, n_2$ and $l_3, m_3, n_3$ as coplanar.
View full solution →If the direction ratios of line $A B$ are $\cos \alpha, \cos \beta, \cos \gamma$, then what will be the direction cosines of line BA?
View full solution →The direction ratios of two mutually perpendicular lines are $1,2,3$ and $3,2, \lambda$. Then write the value of $\lambda$.
View full solution →Find the direction cosines of the line joining the points $(1,0,0)$ and $(0,1,1)$.
View full solution →Show that the line through the point $(1,-1,2)$, $(3,4,-2)$ is perpendicular to the line through the point $(0,3,2)$ and $(3,5,6)$.
View full solution →Find the angle between the pair of lines given by
$\vec{r}=(3 \hat{i}+2 \hat{j}-4 \hat{k})+\lambda(\hat{i}+2 \hat{j}+2 \hat{k})$ and
$\vec{r}=(5 \hat{i}-2 \hat{j})+\mu(3 \hat{i}+2 \hat{j}+6 \hat{k})$.
View full solution →Projections of a line on $x, y, z$ axes are respectively 12,5 and $2 \sqrt{14}$. Find the length of the line segment and its direction cosines.
View full solution →If a line makes angle $\alpha, \beta, \gamma$ with the coordinate axes, then find the value of $\cos 2 \alpha+\cos 2 \beta+$ $\cos 2 \gamma$.
View full solution →Find the direction cosines of a line parallel to the given line: $\frac{4-x}{2}=\frac{y+3}{3}=\frac{z+2}{6}$
View full solution →Find the equation of the line in vector and in Cartesian form that passes through the point with position vector $2 \hat{i}-\hat{j}+4 \hat{k}$ and is in the direction $\hat{i}+2 \hat{j}-\hat{k}$
View full solution →Find the shortest distance between the lines $l_1$ and $l_2$ whose vector equations are
$\vec{r} =\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k})$
$\vec{r} =2 \hat{i}+\hat{j}-\hat{k}+\mu(3 \hat{i}-5 \hat{j}+2 \hat{k})$
View full solution →Find the shortest distance between the following given lines $l_1$ and $l_2$
$\vec{r}=\hat{i}+2 \hat{j}-4 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$
$\text { and } \vec{r}=3 \hat{i}+3 \hat{j}-5 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$
View full solution →From point $P(1,2,3)$ perpendicular $P N$ is drawn to the line $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$. Then find the following :
(i) Coordinates of point N
(ii) Length of PN
View full solution →If the direction cosines of any variable line in its two adjacent position are $l , m , n$ and $l +\delta, m +\delta m$, $n +\delta n$ the angle between those position be $\delta \theta$, then prove that
$
(\delta \theta)^2=(\delta l)^2+(\delta m)^2+(\delta n)^2
$
View full solution →Show that the lines $\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}$ and $\frac{x-2}{1}=\frac{y+4}{3}=\frac{z-6}{5}$ intersect each other. Find also the coordinates of the point of intersection.
View full solution →If the coordinates of the points $A , B , C$ and D are respectively $(1,2,3),(4,5,7),(-4,3,-6)$ and $(2,9,2)$, then the angle between the lines AB and CD will be an acute angle ____________ .
View full solution →A line passing through origin and parallel to z-axis is _________ .
The equations of a line passing through origin and parallel to $x$-axis are _________ .
View full solution →The condition of parallelism of two lines is __________ .
If the direction cosines of a line are $l, m, n$, then the equations of the line are _____________ .
View full solution →