Embibe Experts Solutions for Exercise 5: Assignment Set-1

If the position vector of a point $(-4,-3)$ be $\overrightarrow{a}$, find $\left|\overrightarrow{a}\right|.$

Find the area of a parallelogram, if the diagonals of a parallelogram are represented by the vectors ${\overrightarrow{a}}_1=3\hat{i}+\hat{j}-2\hat{k}$ and ${\overrightarrow{a}}_2=\hat{i}-3\hat{j}+4\hat{k}$

If $\left|\overrightarrow{a}\right|=\sqrt{26},\left|\overrightarrow{b}\right|=7$ and $\left|\overrightarrow{a}\times \overrightarrow{b}\right|=35$, then find the value of $\overrightarrow{a}·\overrightarrow{b}.$

If the vectors $\overrightarrow{a}=3\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{b}=\hat{i}+\lambda \hat{j}-3\hat{k}$ are perpendicular to each other, then find the value of $\lambda .$

Find the area of a triangle, if two adjacent sides of a triangle are represented by the vectors $\overrightarrow{a}=3\hat{i}+4\hat{j}$ and $\overrightarrow{b}=-5\hat{i}+7\hat{j}.$

Let $A$ and $B$ are two points with position vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ respectively, and let $P$ be a point dividing $AB$ internally in the ratio $m:n$. Let $\overrightarrow{OP}=\overrightarrow{r}$, then prove that $\overrightarrow{r}=\frac{m\overrightarrow{b}+n\overrightarrow{a}}{m+n}.$

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-zero vectors. Prove that $\overrightarrow{a}\perp \overrightarrow{b}\iff \left|\overrightarrow{a}+\overrightarrow{b}\right|=\left|\overrightarrow{a}-\overrightarrow{b}\right|$

Using vectors, find the area of the triangle whose vertices are $A(1,1,1), B(1,2,3)$ and $C(2,3,1).$