Question

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.

Answer 1

$\overrightarrow{a}=\overrightarrow{i}-\overrightarrow{j}+7\overrightarrow{k}$

$\overrightarrow{b}=5\overrightarrow{i}-\overrightarrow{j}+\lambda \overrightarrow{k}$

$\overrightarrow{a}+\overrightarrow{b}=6\overrightarrow{i}-2\overrightarrow{j}+(7+\lambda )\overrightarrow{k}$

$\overrightarrow{a}-\overrightarrow{b}=-4\overrightarrow{i}-0\overrightarrow{j}+(7-\lambda )\overrightarrow{k}$

Given $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$ are perpendicular, then

$\overrightarrow{a}+\overrightarrow{b}.\overrightarrow{a}-\overrightarrow{b}=0$

$(6\overrightarrow{i}-2\overrightarrow{j}+(7+\lambda \left)\overrightarrow{k}\right)\times (-4\overrightarrow{i}-0\overrightarrow{j}+(7-\lambda \left)\overrightarrow{k}\right)=0$

$6(-4)+0(-2)+(7+\lambda )(7-\lambda )=0$

$-24+49-{\lambda }^2=0$

${\lambda }^2=25$

Squaring on both the sides, we get

$\lambda =\pm 5$