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 →Perpendicular distance of the point $P (x, y, z)$ from $z$ axis is :
Answer: B.
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
$
\begin{aligned}
\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})
\end{aligned}
$
View full solution →Find the shortest distance between the following given lines $l_1$ and $l_2$
$
\begin{array}{l}
\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})
\end{array}
$
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 →Find the shortest distance between the following lines:
$
\begin{array}{l}
\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k}) \\
\vec{r}=(2 \hat{i}+4 \hat{j}+5 \hat{k})+\mu(4 \hat{i}+6 \hat{j}+8 \hat{k})
\end{array}
$
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 →