Embibe Experts Solutions for Chapter: Three Dimensional Geometry, Exercise 1: Kerala Board-2018

Write all the direction cosines of $x-$axis.

If a line makes angles $\alpha , \beta , \gamma$ with $x, y, z$ axes respectively, then prove that ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =2$

If a line makes equal angles with the three coordinate axes, find the direction cosines of the lines.

Find the equation of the plane through the intersection of the planes $3x-y+2z-4=0 \mathrm{and} x+y+z-2=0$ and the point $\left(2,2,1\right)$.

The Cartesian equation of two lines are given by $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$. Write then vector equation of these two lines.

Find the shortest distance between the lines $\left(\hat{i}-\hat{j}-\hat{k}\right)+\lambda \left(7\hat{i}-6\hat{j}+\hat{k}\right)$ and $\left(3\hat{i}+5\hat{j}+7\hat{k}\right)+\mu \left(\hat{i}-2\hat{j}+\hat{k}\right)$.

Find the distance of the plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ from the origin.

Find the vector and Cartesian equation of the plane passing through $\left(1,0,-2\right)$ and normal to the plane is $i+j-k$.