Maharashtra Board Solutions for Chapter: Vectors, Exercise 4: Exercise 5.4

Find $|\overline{u}\times \overline{v}|$ if $\left|\overline{u}\right|=10,\left|\overrightarrow{v}\right|=2, \overline{u}·\overline{v}=12$

If $\overline{a},\overline{b},\overline{c}$ and $\overline{d}$ are four distinct vectors such that $\overline{a}\times \overline{b}=\overline{c}\times \overline{d}$ and $\overline{a}\times \overline{c}=\overline{b}\times \overline{d}$, prove that $\overline{a}-\overline{d}$ is parallel to $\overline{b}-\overline{c}$

If $\overline{a}=\hat{i}+\hat{j}+\hat{k}$ and $\overline{c}=\hat{j}-\hat{k}$, find a vector $\overline{b}$ satisfying $\overline{a}\times \overline{b}=\overline{c}$ and $\overline{a}·\overline{b}=3$

Find $\overline{a}$, if $\overline{a}\times \hat{i}+2\overline{a}-5\hat{j}=0$.

If $|\overline{a}·\overline{b}|=|\overline{a}\times \overline{b}|$ and $\overline{a}·\overline{b}<0$, then find the angle between $\overline{a}$ and $\overline{b}$

Prove by vector method that $\sin (\alpha +\beta )=\sin a\cos \beta +\cos \alpha \sin \beta$.

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are

$1,3,2$ and $-1,1,2$

If $A(1,2,3)$ and $B(4,5,6)$ are two points, then find the foot of the perpendicular from the point $B$ to the line joining the origin and point $A$