If \(\overrightarrow { a } =\hat { i } -\hat { j } +7\hat { k } \) and \(\overrightarrow { b } =5\hat { i } -\hat { j } +\lambda \hat { k } \) then find the value of \(\lambda\)  so that \(\overrightarrow { a } +\overrightarrow { b } \ and\ \overrightarrow { a } -\overrightarrow { b } \) are perpendicular vectors.

Find a vector \(\overrightarrow { r } \) of magnitude 3\(\sqrt2\) units which makes an angle of \(\pi\over4\) and \(\pi\over2\) with y and z-axis respectively. 

If \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } \ and\ \overset { \rightarrow }{ c } \) are perpendicular to each other, show that 

If a vector \(\vec{r}\) has magnitude 14and direction ratios 2, 3 and - 6. Then, find the direction cosines and components of \(\vec{r}\)given that \(\vec{r}\) makes an acute angle with x-axis.

If the three vectors \(\vec{a}, \vec{b} \text { and } \vec{c}\) are given as \(a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}, b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k} \text { and } c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}\) Then, show that \(\vec{a} \times(\vec{b}+\vec{c})=(\vec{a} \times \vec{b})+(\vec{a} \times \vec{c})\).