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

Let $A B C D$ is parallelogram where $A=(1,2,4), B=(8,7,9)$ and $D=(6,1,5)$. Find direction cosines of line $AC$

Find the projection of the line segment joining $\left(2,-1,3\right)$ and $\left(4,2,5\right)$ on a line which makes equal acute angles with coordinate axes.

Find the coordinates of the point when the line through $(3,2,5)$ and$(-2,3,-5)$ crosses the $x y$ plane.

$\left(ii\right)$ Find the vector form of the equation of a line whose cartesian form is given by

$\frac{2x-4}{1}=\frac{3y+6}{2}=\frac{-6z+6}{1}$

Show that equation of angle bisectors of line $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$ and $\overrightarrow{r}=\overrightarrow{b}+\mu \overrightarrow{a}$ are $\overrightarrow{r}=(\overrightarrow{a}+\overrightarrow{b})+\gamma \left(\right|\overrightarrow{b}|\overrightarrow{a}\pm |\overrightarrow{a}\left|\overrightarrow{b}\right)$

Find equation of plane $\left(ii\right)$ Which passes through $\left(0,1,0\right)$ and contains two vectors $\hat{i}+\hat{j}-\hat{k}$ & $2\hat{i}-\hat{j}.$

Find the equation of the plane passing through the points $\left(3,4,1\right)$ and $\left(0,1,0\right)$ and parallel to the line $\frac{x+3}{2}=\frac{y-3}{7}=\frac{z-2}{5}$$.$

The locus represented by $x y+y z=0$ is