#### Need solution for rd sharma maths class 12 chapter 24 vector or cross product Exercises Multiple choice questions question 2 maths textbook

Answer: $\overrightarrow{b}=\overrightarrow{c}$

Given: $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{a}.\overrightarrow{c}$ and $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{c},\overrightarrow{a}\neq 0$

Hint: You must know about dot product and cross product.

Explanation:

$\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{a}.\overrightarrow{c}$                                        ………………. (1)

$\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{c}$                                 ………………. (2)

Squaring and adding (1) and (2)

$\begin{gathered} (\vec{a} \cdot \vec{b})^{2}+|\vec{a} \times \vec{b}|^{2}=(\vec{a} \cdot \vec{c})^{2}+|\vec{a} \times \vec{c}|^{2} \\ \end{gathered}$

$\Rightarrow|\vec{a}|^{2}|\vec{b}|^{2} \cos ^{2} \theta+|\vec{a}|^{2}|\vec{b}|^{2} \sin ^{2} \theta=|\vec{a}|^{2}|\vec{c}|^{2} \cos ^{2} \theta+|\vec{a}|^{2}|\vec{c}|^{2} \sin ^{2} \theta \quad\left[\begin{array}{l} \because \vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta \\ \vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \sin \theta \end{array}\right]$

\begin{aligned} &\Rightarrow|\vec{a}|^{2}|\vec{b}|^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=|\vec{a}|^{2}|\vec{c}|^{2}\left(\sin ^{2} \theta+\cos ^{2} \theta\right) \\ &\Rightarrow|\vec{a}|^{2}|\vec{b}|^{2}(1)=|\vec{a}|^{2}|\vec{c}|^{2}(1) \end{aligned} \quad\left[\because \sin ^{2} \theta+\cos ^{2} \theta=1\right]

\begin{aligned} &\Rightarrow|\vec{a}|^{2}|\vec{b}|^{2}=|\vec{a}|^{2}|\vec{c}|^{2} \\ &\Rightarrow|\vec{b}|^{2}=|\vec{c}|^{2} \\ &\Rightarrow|\vec{b}|=|\vec{c}| \\ &\Rightarrow \vec{b}=\vec{c} \end{aligned}