EXERCISE

If $ABC$ is a triangle (shown in the adjoining figure), then  $\overrightarrow{AB}+\overrightarrow{BC}-\overrightarrow{CA}=0$

If $\overrightarrow{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\overrightarrow{b}=2\hat{i}+4\hat{j}+9\hat{k}$, then find a unit vector parallel to $\overrightarrow{a}+\overrightarrow{b}$.

Find a vector in the direction of the vector $\overrightarrow{a}=\hat{i}-2\hat{j}$, whose magnitude is $7$ units. 

Write a vector of magnitude $9$ units in the direction of the vector $-2\hat{i}+\hat{j}+2\hat{k}$.

Write the value of $p$ for which $\overrightarrow{a}=3\hat{i}+2\hat{j}+9\hat{k}$ and $\overrightarrow{b}=\hat{i}+p\hat{j}+3\hat{k}$ are parallel vectors. 

What is the cosine of the angle which the vector $\sqrt{2}\hat{i}+\hat{j}+\hat{k}$ makes with $y$-axis?

The vectors $\overrightarrow{a}$  and $\overrightarrow{b}$ are non-zero non-collinear. For what value of $x$, the vectors $(x-2)\overrightarrow{a}+\overrightarrow{b}$ and $(2x+1)\overrightarrow{a}-\overrightarrow{b}$​​​​​are collinear ?

Using vectors, find the value of $\lambda$ such that the points $\left(\lambda , -10, 3\right), \left(1, -1, 3\right)$ and $\left(3, 5, 3\right)$ are collinear.