Consider the lines L_1: frac{x+1}{3}=frac{y+2 | vedclass

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

Question

(B) The lines $L_1$ and $L_2$ are parallel to the vectors $\vec{b}_1 = 3\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b}_2 = \hat{i} + 2\hat{j} + 3\hat{k}$

Vedclass · Accepted Answer

$\frac{-\hat{i}-7 \hat{j}+5 \hat{k}}{5 \sqrt{3}}$

Vedclass · Answer

(B) The lines $L_1$ and $L_2$ are parallel to the vectors $\vec{b}_1 = 3\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b}_2 = \hat{i} + 2\hat{j} + 3\hat{k}$ respectively.The unit vector perpendicular to both $L_1$ and $L_2$ is given by $\hat{n} = \pm \frac{\vec{b}_1 	imes \vec{b}_2}{|\vec{b}_1 	imes \vec{b}_2|}$.First,calculate the cross product $\vec{b}_1 	imes \vec{b}_2$:$\vec{b}_1 	imes \vec{b}_2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & 1 & 2 \ 1 & 2 & 3 \end{vmatrix} = \hat{i}(3-4) - \hat{j}(9-2) + \hat{k}(6-1) = -\hat{i} - 7\hat{j} + 5\hat{k}$.Next,calculate the magnitude $|\vec{b}_1 	imes \vec{b}_2|$:$|\vec{b}_1 	imes \vec{b}_2| = \sqrt{(-1)^2 + (-7)^2 + 5^2} = \sqrt{1 + 49 + 25} = \sqrt{75} = 5\sqrt{3}$.Thus,the unit vector is $\hat{n} = \frac{-\hat{i} - 7\hat{j} + 5\hat{k}}{5\sqrt{3}}$.

Consider the lines $L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}$ and $L_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}$. Then the unit vector perpendicular to both $L_1$ and $L_2$ is:

Similar Questions

If $(2 \hat{i} + 6 \hat{j} + 27 \hat{k}) \times (\hat{i} + \lambda \hat{j} + \mu \hat{k}) = 0$,then $\lambda + \mu =$ . . . . . . .

If the vectors $\hat{i}-3 \hat{j}+2 \hat{k}$ and $-\hat{i}+2 \hat{j}$ represent the diagonals of a parallelogram,then its area will be

If $\vec{a}=2 \hat{i}+\hat{j}-3 \hat{k}$,$\vec{b}=\hat{i}-2 \hat{j}+\hat{k}$,$\vec{c}=-\hat{i}+\hat{j}-4 \hat{k}$ and $\vec{d}=\hat{i}+\hat{j}+\hat{k}$,then $|(\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})|=$

The area of the quadrilateral $ABCD$ with vertices $A(0,4,1)$,$B(2,3,-1)$,$C(4,5,0)$,and $D(2,6,2)$ is equal to

The unit vector perpendicular to the vectors $6i + 2j + 3k$ and $3i - 6j - 2k$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module