(3, 0, 2) and (0, 2, k) are the direction rat | vedclass

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

Question

(B) Let the direction ratios of the two lines be $\vec{a} = (3, 0, 2)$ and $\vec{b} = (0, 2, k)$.The formula for the cosine of the angle $	heta$ betw

Vedclass · Accepted Answer

$\pm 3$

Vedclass · Answer

(B) Let the direction ratios of the two lines be $\vec{a} = (3, 0, 2)$ and $\vec{b} = (0, 2, k)$.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 $|\cos 	heta| = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$.Substituting the given values: $|\cos 	heta| = \frac{|(3)(0) + (0)(2) + (2)(k)|}{\sqrt{3^2 + 0^2 + 2^2} \sqrt{0^2 + 2^2 + k^2}} = \frac{|2k|}{\sqrt{13} \sqrt{4 + k^2}}$.Given $|\cos 	heta| = \frac{6}{13}$, we have $\frac{6}{13} = \frac{|2k|}{\sqrt{13} \sqrt{4 + k^2}}$.Squaring both sides: $\frac{36}{169} = \frac{4k^2}{13(4 + k^2)}$.Simplifying: $\frac{9}{13} = \frac{k^2}{4 + k^2}$.$9(4 + k^2) = 13k^2 \Rightarrow 36 + 9k^2 = 13k^2 \Rightarrow 4k^2 = 36 \Rightarrow k^2 = 9$.Thus, $k = \pm 3$.

$(3, 0, 2)$ and $(0, 2, k)$ are the direction ratios of two lines and $\theta$ is the angle between them. If $|\cos \theta| = \frac{6}{13}$, then $k =$

Similar Questions

If non-zero vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ are related by $\vec{a} = 8\vec{b}$ and $\vec{c} = -7\vec{b}$,find the angle between $\vec{a}$ and $\vec{c}$.

The vectors $a = 2\hat{i} + 3\hat{j} + 6\hat{k}$ and $b$ are collinear and $|b| = 21$,then $b =$

If $C$ is the mid-point of the line segment $AB$ and $P$ is any point outside the line $AB$,then

If $\bar{a}$ and $\bar{b}$ are two non-parallel unit vectors and the vector $\alpha \bar{a} + \bar{b}$ bisects the internal angle between $\bar{a}$ and $\bar{b}$,then $\alpha$ is equal to

Classify the following measure as a scalar or a vector:
$20 \text{ m/s}$ towards north

Vedclass Test Series

Exam Paper Generator

Online Exam Module