Let $\bar{a}=3 \hat{i}+2 \hat{j}+x \hat{k}$ and $\bar{b}=\hat{i}-\hat{j}+\widehat{k}$, for some real $x$. Then $|\bar{a} imes \bar{b}|=r$ is possible, if

Question

Let $\bar{a}=3 \hat{i}+2 \hat{j}+x \hat{k}$ and $\bar{b}=\hat{i}-\hat{j}+\widehat{k}$, for some real $x$. Then $|\bar{a} 	imes \bar{b}|=r$ is possible, if

Aicharya · Accepted Answer

$\begin{aligned} & \vec{a} 	imes \vec{b}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \ 3 & 2 & x \ 1 & -1 & 1\end{array}ight|=(2+x) \hat{i}+(x-3) \hat{j}+(-3-2) \hat{k} \ & 	ext { now } r=|\vec{a} 	imes \vec{b}|=\sqrt{(2+x)^2+(x-3)^2+(-5)^2} \ & =\sqrt{2 x^2-2 x+38} \ & =\sqrt{2\left(x-\frac{1}{2}ight)^2+\frac{75}{2}} \ & \Rightarrow r_{\min }=\sqrt{\frac{75}{2}} 	ext { at } x=\frac{1}{2} \ & \Rightarrow r \geq 5 \cdot \sqrt{\frac{3}{2}}\end{aligned}$

Let $\bar{a}=3 \hat{i}+2 \hat{j}+x \hat{k}$ and $\bar{b}=\hat{i}-\hat{j}+\widehat{k}$, for some real $x$.…

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