Embibe Experts Solutions for Chapter: Vector & Three Dimensional Geometry, Exercise 1: EXERCISE

The direction ratios of a normal to the plane through $(1,0,0)$ and $(0,1,0),$ which makes an angle of $\frac{\mathrm{\pi }}{4}$ with the plane $x+y=3$ are

From the point $P(a,b,c),$ let perpendiculars $PL$ and $PM$ be drawn to $YOZ$ and $ZOX$ planes, respectively. Then the equation of the plane $OLM$ is

The line $\frac{x+6}{5}=\frac{y+10}{3}=\frac{z+14}{8}$ is the hypotenuse of an isosceles right-angled triangle whose opposite vertex is $(7,2,4).$ Then which of the following is not the side of the triangle

The ratio in which the line segment joining the points whose position vectors are $2\hat{i}-4\hat{j}-7\hat{k}$ and $-3\hat{i}+5\hat{j}-8\hat{k}$ is divided by the plane whose equation is $\hat{r}·(\hat{i}-2\hat{j}+3\hat{k})=13$ is

$L_1$ and $L_2$ are two lines whose vector equations are $L_1:\overrightarrow{r}=\lambda \left(\left(\cos \theta +\sqrt{3}\right)\hat{i}+\left(\sqrt{2}\sin \theta \right)\hat{j}+\left(\cos \theta -\sqrt{3}\right)\hat{k}\right)$

$L_2:\overrightarrow{r}=\mu (a\hat{i}+b\hat{j}+c\hat{k}),$ where $\lambda$ and $\mu$ are scalars and $\alpha$ is the acute angle between $L,$ and $L_2.$ If the angle

$\text{'}\alpha \text{'}$ is independent of $\theta ,$ then the value of $\text{'}\alpha \text{'}$ is

Equation of the plane passing through the points $(2,2,1)$ and $(9,3,6),$ and$\perp$ to the plane $2x+6y+6z-1$$=0$ is

What is the equation of the plane which passes through the $z$-axis and is perpendicular to the line $\frac{x-a}{\cos \theta }=\frac{y-2}{\sin \theta }=\frac{z-3}{0}?$

A straight line $L$ on the $xy$ -plane bisects the angle between $OX$ and $OY.$ What are the direction cosines of $L$?