Let overline{a}=2 hat{i}+hat{j}-2 hat{k} and | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given $|\overline{c}-\overline{a}|=2 \sqrt{2}$. Squaring both sides,we get $|\overline{c}|^2+|\overline{a}|^2-2(\overline{a} \cdot \overline{c})=8

Vedclass · Accepted Answer

$\frac{3\sqrt{3}}{2}$

Vedclass · Answer

(B) Given $|\overline{c}-\overline{a}|=2 \sqrt{2}$. Squaring both sides,we get $|\overline{c}|^2+|\overline{a}|^2-2(\overline{a} \cdot \overline{c})=8$. Since $|\overline{a}| = \sqrt{2^2+1^2+(-2)^2} = \sqrt{9} = 3$ and $\overline{a} \cdot \overline{c}=|\overline{c}|$,we have $|\overline{c}|^2+9-2|\overline{c}|=8$. This simplifies to $|\overline{c}|^2-2|\overline{c}|+1=0$,which is $(|\overline{c}|-1)^2=0$,so $|\overline{c}|=1$. Next,calculate $\overline{a} 	imes \overline{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & 1 & -2 \ 1 & 1 & 0 \end{vmatrix} = \hat{i}(0 - (-2)) - \hat{j}(0 - (-2)) + \hat{k}(2 - 1) = 2\hat{i}-2\hat{j}+\hat{k}$. The magnitude is $|\overline{a} 	imes \overline{b}| = \sqrt{2^2+(-2)^2+1^2} = \sqrt{9} = 3$. Finally,$|(\overline{a} 	imes \overline{b}) 	imes \overline{c}| = |\overline{a} 	imes \overline{b}| |\overline{c}| \sin\left(\frac{2\pi}{3}ight) = 3 	imes 1 	imes \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}$.

Similar Questions

If $3\hat{i} + 4\hat{j}$ and $-5\hat{i} + 7\hat{j}$ are the vector sides of a triangle,then its area is given by

Find $|\vec{a} \times \vec{b}|,$ if $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$ and $\vec{b}=3 \hat{i}+5 \hat{j}-2 \hat{k}$.

If $a = i + j + 2k$ and $b = 3i + j + k$,then the value of $a \times b$ is:

Let $\bar{a}$ and $\bar{b}$ be two vectors such that $|\bar{a}|=1$,$|\bar{b}|=4$,and $\bar{a} \cdot \bar{b}=2$. If $\bar{c}=(2 \bar{a} \times \bar{b})-3 \bar{b}$,then the angle between $\bar{b}$ and $\bar{c}$ is

If $\theta$ is the angle between the vectors $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} + \hat{k}$,find the value of $\sin \theta$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module