The angle between the line frac{x - 1}{-2} = | vedclass

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

Question

(A) The direction vector of the line is $\vec{b} = -2\hat{i} + \hat{j} + 2\hat{k}$.The normal vector to the plane is $\vec{n} = 3\hat{i} + 2\hat{j} +

Vedclass · Accepted Answer

$\sin^{-1}\left(\frac{8}{21}\right)$

Vedclass · Answer

(A) The direction vector of the line is $\vec{b} = -2\hat{i} + \hat{j} + 2\hat{k}$.The normal vector to the plane is $\vec{n} = 3\hat{i} + 2\hat{j} + 6\hat{k}$.The angle $	heta$ between a line and a plane is given by $\sin 	heta = \frac{|\vec{b} \cdot \vec{n}|}{|\vec{b}| |\vec{n}|}$.Calculate the dot product: $\vec{b} \cdot \vec{n} = (-2)(3) + (1)(2) + (2)(6) = -6 + 2 + 12 = 8$.Calculate the magnitudes: $|\vec{b}| = \sqrt{(-2)^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3$.$|\vec{n}| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7$.Thus,$\sin 	heta = \frac{|8|}{3 	imes 7} = \frac{8}{21}$.Therefore,$	heta = \sin^{-1}\left(\frac{8}{21}ight)$.

The angle between the line $\frac{x - 1}{-2} = \frac{y - 2}{1} = \frac{z + 1}{2}$ and the plane $3x + 2y + 6z = 1$ is:

Similar Questions

The equation of the plane passing through the intersection of the planes $x+y+z=1$ and $2x+3y-z+4=0$ and parallel to the $X$-axis is

The value of $k$ such that the line $\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}$ lies in the plane $2x-4y+z=7$ is:

The equation of the line passing through $(-4, 1, 3)$,parallel to the plane $x + 2y - z - 5 = 0$ and intersecting the line $\frac{x + 1}{-3} = \frac{y - 3}{2} = \frac{z - 2}{-1}$ is

The distance of the point $(1, 6, 2)$ from the point of intersection of the line $\frac{x-2}{3} = \frac{y+1}{4} = \frac{z-2}{12}$ and the plane $x-y+z=16$ is (in $\text{ units}$)

Let $A(2,5,7)$ be the image of the point $B(1,-2,3)$ with respect to a plane $\pi$. Let $C$ be the point where $AB$ meets the plane $\pi$. Let $D=(2,1,6)$. Then the direction cosines of $CD$ are

Vedclass Test Series

Exam Paper Generator

Online Exam Module