Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 1: CG PET 2019

If $\overrightarrow{a}=2\hat{\mathrm{i}}+5\hat{\mathrm{j}}$ and $\overrightarrow{b}=2\hat{\mathrm{i}}-\hat{\mathrm{j}}$, then the unit vector along $\overrightarrow{a}+\overrightarrow{b}$ will be

${\left(\overrightarrow{r}·\hat{\mathrm{i}}\right)}^2+{\left(\overrightarrow{r}·\hat{\mathrm{j}}\right)}^2+{\left(\overrightarrow{r}·\hat{\mathrm{k}}\right)}^2$ is equal to

The component of $\hat{\mathrm{i}}+\hat{\mathrm{j}}$ along $\hat{\mathrm{j}}+\hat{\mathrm{k}}$ will be

The vector $2\hat{i}+\hat{j}-\hat{k}$ is perpendicular to $\hat{i}-4\hat{j}+\lambda \hat{k}$, then $\lambda$ is equal to

For any vectors, $\overrightarrow{a}, \overrightarrow{b} \& \overrightarrow{c}$, $\overrightarrow{a}\times \left(\overrightarrow{b}+\overrightarrow{c}\right)+\overrightarrow{b}\times \left(\overrightarrow{c}+\overrightarrow{a}\right)+\overrightarrow{c}\times \left(\overrightarrow{a}+\overrightarrow{b}\right)=$

If $\overrightarrow{u}=\hat{i}\times \left(a\times \hat{i}\right)+\hat{j}\times \left(a\times \hat{j}\right)+\hat{k}\times \left(a\times \hat{k}\right),$ then

If $\left|\overrightarrow{a}\right|=1,\left|\overrightarrow{b}\right|=2$ and $\left|\overrightarrow{c}\right|=3$, then $[\overrightarrow{a}-\overrightarrow{b} \overrightarrow{b}-\overrightarrow{c} \overrightarrow{c}-\overrightarrow{a}]=$

If the volume of a tetrahedron having vertices with position vectors $\hat{i}-6\hat{j}+10\hat{k},-\hat{i}-3\hat{j}+7\hat{k},5\hat{i}-\hat{j}+\lambda \hat{k}$ and $7\hat{i}-4\hat{j}+7\hat{k}$ is $11 \mathrm{units}$, then $\lambda =$