If the angle between the lines whose directio | vedclass

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

Question

(C) The formula for the cosine of the angle $	heta$ between two lines with direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ is given by $\cos 	h

Vedclass · Accepted Answer

$\pm 5$

Vedclass · Answer

(C) The formula for the cosine of the angle $	heta$ between two lines with direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ is given by $\cos 	heta = \left| \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} ight|$.Given $	heta = \frac{\pi}{3}$,$a_1=4, b_1=-3, c_1=5$ and $a_2=3, b_2=4, c_2=k$.Substituting these values,we get $\cos \frac{\pi}{3} = \left| \frac{4(3) + (-3)(4) + 5k}{\sqrt{4^2 + (-3)^2 + 5^2} \sqrt{3^2 + 4^2 + k^2}} ight|$.$\frac{1}{2} = \left| \frac{12 - 12 + 5k}{\sqrt{16 + 9 + 25} \sqrt{9 + 16 + k^2}} ight|$.$\frac{1}{2} = \left| \frac{5k}{\sqrt{50} \sqrt{25 + k^2}} ight| = \left| \frac{5k}{5\sqrt{2} \sqrt{25 + k^2}} ight| = \left| \frac{k}{\sqrt{2} \sqrt{25 + k^2}} ight|$.Squaring both sides,$\frac{1}{4} = \frac{k^2}{2(25 + k^2)}$.$2(25 + k^2) = 4k^2$.$50 + 2k^2 = 4k^2$.$2k^2 = 50 \Rightarrow k^2 = 25$.Therefore,$k = \pm 5$.

If the angle between the lines whose direction ratios are $4, -3, 5$ and $3, 4, k$ is $\frac{\pi}{3}$,then $k=$

Similar Questions

Find the projection of the line segment joining the points $P(7, -5, 11)$ and $Q(-2, 8, 13)$ on a line $AB$ having direction cosines $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$.

$A$ line makes angles $60^{\circ}, 45^{\circ}, \theta$ with positive $X, Y, Z$-axes respectively. If $\theta$ is an acute angle, then $\tan \theta=$

Let $P = (x_1, y_1, z_1)$ and $Q = (x_2, y_2, z_2)$ be two points. If the direction cosines of a line $AB$ are $l, m, n$,then the projection of the line segment $PQ$ on the line $AB$ is:

The acute angle between two lines such that the direction cosines $l, m, n$ of each of them satisfy the equations $l+m+n=0$ and $l^2+m^2-n^2=0$ is ............ $^o$

If the direction cosines of two lines are such that $l+m+n=0$ and $l^2+m^2-n^2=0$, then the angle between them is

Vedclass Test Series

Exam Paper Generator

Online Exam Module