Maharashtra Board Solutions for Chapter: Vectors, Exercise 5: Exercise 5.5

If the vectors $-3\hat{i}+4\hat{j}-2\hat{k},\hat{i}+2\hat{k}$ and $\hat{i}-\widehat{pj}$ are coplanar, then find the value of $p$.

Prove that :

$(\overline{a}+2\overline{b}-\overline{c})\left[\right(\overline{a}-\overline{b})\times \overline{a}-\overline{b}-\overline{c}]=3[\overline{a} \overline{b} \overline{c}]$

 If $\overline{c}=3\overline{a}-2\overline{b}$  and $\left[\begin{array}{ll}\overline{a}& \overline{b}+\overline{c} \overline{a}+\overline{b}+\overline{c}\end{array}\right]=0$ then prove that $\left[\overline{a}\overline{b}\overline{c}\right]=0$

If $u=\hat{i}-2\hat{j}+\hat{k}, \overline{r}=3\hat{i}+\hat{k}$ and $\overline{w}=\hat{j}-\hat{k}$ are given vectors, then find $[\overline{u}+\overline{w}]·\left[\right(\overline{u}\times \overline{r})\times (\overline{r}\times \overline{w}\left)\right]$.

Find the volume of a tetrahedron whose vertices are $A(-1,2,3) B(3,-2,1), C(2,1,3)$ and $D(-1,-2,4)$

If $\overline{a}=\hat{i}+2\hat{j}+3\hat{k}, \overline{b}=3\hat{i}+2\hat{j}$ and $c=2\hat{i}+\hat{j}+3\hat{k}$ then verify that $\overline{a}\times (\overline{b}\times \overline{c})=(\overline{a}·\overline{c})\overline{b}-(\overline{a}·\overline{b})\overline{c}$

If, $\overline{a}=\hat{i}-2\hat{j}, \overline{b}=\hat{i}+2\hat{j}$ and$\overline{c}=2\hat{i}+\hat{j}-2$ then find

(i) $\overline{a}\times (\overline{b}\times \overline{c})$  (ii) $\left(\overline{a}\times \overline{b}\right)\times \overline{c}$ 

Are the results same? Justify.

Show that $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})+\overrightarrow{b}\times (\overrightarrow{c}\times \overrightarrow{a})+\overrightarrow{c}\times (\overrightarrow{a}\times \overrightarrow{b})=0$