Perpendicular Distance of a Point from a Plane

Q.

A straight line through the origin $O$ meets the parallel lines $4x+2y=9$ and $2x+y+6=0$ at points $P$ and $Q$ respectively. Then the point $O$ divides the segment $PQ$ in the ratio

1. $1:2$

2. $3:4$

3. $2:1$

4. $4:3$

Q. If $Q(0,-1,-3)$ is the image of the point $P$ in the plane $3x-y+4z=2$ and $R$ is the point $(3,-1,-2),$ then the area (in sq. units) of $\Delta PQR$ is :

1. $\frac{\sqrt{91}}{4}$
2. $\frac{\sqrt{65}}{2}$
3. $2\sqrt{13}$
4. $\frac{\sqrt{91}}{2}$

Q.

If $A=\{a,b\},B=\{c,d\},C=\{d,e\}$, then $\left\{\right(a,c),(a,d),(a,e),(b,c),(b,d),(b,e\left)\right\}$ is equal to

1. $A\cap (B\cup C)$

2. $A\cup (B\cap C)$

3. $A\times (B\cup C)$

4. $A\times (B\cap C)$

Q.

The distance of the point $\left(1,-2,3\right)$ from the plane $\mathrm{x}-\mathrm{y}+\mathrm{z}=5$ measured parallel to the line $\left(\frac{x}{2}\right)=\left(\frac{y}{3}\right)=\left(\frac{z}{-6}\right)$ is:

1. $\frac{1}{7}$

2. $7$

3. $\frac{7}{5}$

4. $1$

Q.

The equation of the planes parallel to the plane $x-2y+2z-3=0$ which are at unit distance from the point $\left(1,2,3\right)$ is $ax+by+cz+d=0$. If $\left(b-d\right)=K\left(c-a\right)$, then the positive value of $K$ is

Q.

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).

Q. Suppose that $\overrightarrow{p},\overrightarrow{q}$ and $\overrightarrow{r}$ are three non-coplanar vectors in $R^3.$ Let the components of a vector $\overrightarrow{s}$ along $\overrightarrow{p},\overrightarrow{q},$ and $\overrightarrow{r}$ be $4,3,$ and $5,$ respectively. If the components of this vector $\overrightarrow{s}$ along $(-\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r}),$ $(\overrightarrow{p}-\overrightarrow{q}+\overrightarrow{r}),$ and $(-\overrightarrow{p}-\overrightarrow{q}+\overrightarrow{r}),$ are $x,y,$ and $z,$ respectively, then the value of $2x+y+z$ is

Q.

A point from a vector starts is called ...... and where it ends is called its ......

1. initial point, end point

2. initial point, end point

3. terminal point, end point.

4. Origin, end point

Q.

Two vectors $\stackrel{\rightharpoonup }{A}$and $\stackrel{\rightharpoonup }{B}$ are such that $\stackrel{\rightharpoonup }{A}+\stackrel{\rightharpoonup }{B}=\stackrel{\rightharpoonup }{A}-\stackrel{\rightharpoonup }{B}$. Then

1. $\stackrel{\rightharpoonup }{A}.\stackrel{\rightharpoonup }{B}=0$

2. $\stackrel{\rightharpoonup }{A}\times \stackrel{\rightharpoonup }{B}=0$

3. $\stackrel{\rightharpoonup }{A}=0$

4. $\stackrel{\rightharpoonup }{B}=0$

Q. Explain: Shifting of origin, coordinates of the point dividing a line segment externally

Q.

If $a$ is perpendicular to both $b$ and $c$, then?

1. $a+(b+c)=0$

2. $a\times (b+c)=0$

3. $a\times (b\times c)=0$

4. $a·(b\times c)=0$

Q.

Find the equation of the plane passing through the line of intersection of the planes $r.(\hat{i}+\hat{j}+\hat{k})=1andr.(2\hat{i}+3\hat{j}-\hat{k})+4=0$ and parallel to X-axis.

Q. Write the distance of the point P(2, 3, 5) from the xy-plane.

Q.

Translate the triangle 4 units’ right and 3 units down. What are the coordinates of the image?

Q. Show that the four points A, B, C and D with position vectors $4\hat{i}+5\hat{j}+\hat{k},-\hat{j}-\hat{k},3\hat{i}+9\hat{j}+4\hat{k}\text{and}4(-\hat{i}+\hat{j}+\hat{k})$ respectively are coplanar.

Q.

Translate a point $(\mathrm{x},\mathrm{y})$ $3$ units left and $5$ units up. Then translate the image $5$ units right and $2$ units up. What are the coordinates of the point after the translations?

Q.

The distance of the plane $6\mathrm{x}-3\mathrm{y}+2\mathrm{z}+14=0$ from the origin is

1. $2$

2. $1$

3. $14$

4. $8$

Q.

A point moves in such a way that the sum of its distance from $xy$-plane and $yz$-plane remains equal to its distance from $zx$-plane. The locus of the point is

1. $x-y-z=2$

2. $x+y-z=2$

3. $x-y+z=2$

4. $x-y+z=0$

Q.

The equation of the sphere touching the three co-ordinate planes is
[AMU 2002]

1. 

2. 

3. 

4. None of these

Q. A rod of length $2$ units whose one end is $(1,0,-1)$ and other end touches the plane $x-2y+2z+4=0$. Then

1. the rod sweeps a figure whose volume is $\pi$ cubic units.
2. the area of the figure which the rod traces on the plane is $2\pi$ units.
3. the length of projection of the rod on the plane is $\sqrt{3}$ units.
4. the centre of the region which the rod traces on the plane is $(\frac{2}{3},\frac{2}{3},\frac{-5}{3})$

Q.

If $3$ and $4$ are intercepts of a line $L=0$, then the distance of $L=0$ from the origin is

1. $5$ units

2. $12$ units

3. $\frac{5}{12}$ units

4. $\frac{12}{5}$ units

Q. The volume of the tetrahedron included between the plane 3x + 4y - 5z - 60 = 0 and the coordinate planes is

1. 60
2. 120
3. 720
4. 600

Q.

The perpendicular distance of the point $P(6,7,8)$ from $XY$- plane is

1. $6$

2. $7$

3. $10$

4. $8$

Q. If the volume of the tetrahedron with edges $\hat{i}+\hat{j}+\hat{k},\hat{i}+a\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}-\hat{k}$ is $6$ cubic units, then $a$ is

1. $1$
2. $-1$
3. $2$
4. $-17$

Q. If the points (1, 1, p ) and (−3, 0, 1) be equidistant from the plane , then find the value of p .

Q. If the points $(1,1,\lambda )$ and $(-3,0,1)$ are equidistant from the plane, $3x+4y-12z+13=0,$ then $\lambda$ satisfies the equation:

1. $3x^2-10x+21=0$
2. $3x^2+10x-13=0$
3. $3x^2-10x+7=0$
4. $3x^2+10x+7=0$

Q.

The vector joining the points A (2, – 3, 1) and B (1, – 2, – 5) directed from B to A is:

1. 

2. 

3. 

4. 

Q. The direction angles of the line $x=4z+3,y=2-3z$ are $\alpha ,\beta$ and $\gamma$, then $\cos \alpha +\cos \beta +\cos \gamma =$____

1. $\frac{2}{\sqrt{26}}$
2. $\frac{8}{\sqrt{26}}$
3. $1$
4. $2$

Q. Let $Q$ and $R$ be the images of a point $P(–3,1)$ in the mirror $x^2-y^2-x+y=0$. If $S$ is the area of triangle $PQR,$ then the value of$\left[\sqrt{S}\right]$ is
$($ $\left[k\right]$ denotes the greatest integer less than or equal to $k.)$

Q.

The distance between the points $(0,5)$ and $(-5,0)$ is

1. $5$

2. $5\sqrt{2}$

3. $2\sqrt{5}$

4. $10$