Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 1: Exercise-1

(iii) Let$ABCD$ is trapezium such that $\overrightarrow{AB}=3\overrightarrow{DC}$. $E$ divides line segement $AB$ internally in the ratio $2:1$and $F$ is mid point of $DC$. If position vector of $A,B$ and $C$ are $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ respectively then find vector $\overrightarrow{FE}$.

Show that the points $A, B, C$ with position vectors $2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$ respectively are the vertices of a right angled triangle. Also find the remaining angles of the triangle.

Prove using vectors: If two medians of a triangle are equal in magnitude, then it is isosceles.

$P,Q$ are the mid-points of the non-parallel sides $BC$ and $AD$ of a trapezium $ABCD$. Show that $∆APD=\Delta CQB.$

Show that  $\left\{\right(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\times (\overrightarrow{c}-\overrightarrow{b}\left)\right\}·\overrightarrow{a}=2\left[\begin{array}{lll}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]$.

Let there exist a vector $\overrightarrow{x}$ satisfying the conditions $\overrightarrow{x}\times \overrightarrow{a}=\overrightarrow{c}\times \overrightarrow{d}$ and $\overrightarrow{x}+2\overrightarrow{d}=(\overrightarrow{v}\times \overrightarrow{d}).$ Find $\overrightarrow{x}$ in terms of $\overrightarrow{a},\overrightarrow{c}$ and $\overrightarrow{d}$

Given $\left|\overrightarrow{a}\right|=10,\left|\overrightarrow{b}\right|=2$ and $\overrightarrow{a}·\overrightarrow{b}=12$, then find $|\overrightarrow{a}\times \overrightarrow{b}|$

The value of $\overrightarrow{r}$ if exist where $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$ and $\overrightarrow{r}\times \overrightarrow{c}=\overrightarrow{d}$ is