The vectors are bar{a}=2 hat{i}+hat{j}-2 hat{ | vedclass

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

Question

(B) Given that the angle between $\bar{a} 	imes \bar{b}$ and $\bar{c}$ is $\frac{\pi}{4}$.Using the definition of the cross product magnitude,$|(\bar

Vedclass · Accepted Answer

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

Vedclass · Answer

(B) Given that the angle between $\bar{a} 	imes \bar{b}$ and $\bar{c}$ is $\frac{\pi}{4}$.Using the definition of the cross product magnitude,$|(\bar{a} 	imes \bar{b}) 	imes \bar{c}| = |\bar{a} 	imes \bar{b}| |\bar{c}| \sin(\frac{\pi}{4})$. ... $(i)$First,calculate $\bar{a} 	imes \bar{b}$:$\bar{a} 	imes \bar{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & 1 & -2 \ 1 & 1 & 0 \end{vmatrix} = \hat{i}(0 - (-2)) - \hat{j}(0 - (-2)) + \hat{k}(2 - 1) = 2\hat{i} - 2\hat{j} + \hat{k}$.The magnitude is $|\bar{a} 	imes \bar{b}| = \sqrt{2^2 + (-2)^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3$.Given $|\bar{c} - \bar{a}| = 2\sqrt{2}$. Squaring both sides:$|\bar{c}|^2 + |\bar{a}|^2 - 2(\bar{a} \cdot \bar{c}) = (2\sqrt{2})^2 = 8$.Since $|\bar{a}| = \sqrt{2^2 + 1^2 + (-2)^2} = 3$ and $\bar{a} \cdot \bar{c} = |\bar{c}|$,we have:$|\bar{c}|^2 + 3^2 - 2|\bar{c}| = 8$.$|\bar{c}|^2 - 2|\bar{c}| + 9 = 8 \implies |\bar{c}|^2 - 2|\bar{c}| + 1 = 0$.$(|\bar{c}| - 1)^2 = 0 \implies |\bar{c}| = 1$.Substituting these values into equation $(i)$:$|(\bar{a} 	imes \bar{b}) 	imes \bar{c}| = 3 	imes 1 	imes \sin(\frac{\pi}{4}) = 3 	imes \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}}$.

Similar Questions

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}+\hat{k}$ are two vectors,and $\vec{c}$ is a unit vector lying in the plane of $\vec{a}$ and $\vec{b}$ such that $\vec{c}$ is perpendicular to $\vec{b}$,then find the value of $\vec{c} \cdot (\hat{i}+\hat{j}+2\hat{k})$.

$A$ non-zero vector $a$ is parallel to the line of intersection of the plane determined by the vectors $i, i + j$ and the plane determined by the vectors $i - j, i + k.$ The angle between $a$ and the vector $i - 2j + 2k$ is

The area of the triangle having vertices as $i - 2j + 3k,$ $- 2i + 3j - k,$ and $4i - 7j + 7k$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module