A unit vector hat{a} makes angles frac{pi}{3} | vedclass

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

Question

(A) Let the direction angles of the unit vector $\hat{a}$ be $\alpha, \beta, 	ext{ and } \gamma$.Given $\alpha = \frac{\pi}{3}$,$\beta = \frac{\pi}{4

Vedclass · Accepted Answer

$\frac{2 \pi}{3}$

Vedclass · Answer

(A) Let the direction angles of the unit vector $\hat{a}$ be $\alpha, \beta, 	ext{ and } \gamma$.Given $\alpha = \frac{\pi}{3}$,$\beta = \frac{\pi}{4}$,and $\gamma = 	heta$.We know that for any unit vector,the sum of the squares of the direction cosines is $1$:$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$Substituting the given values:$\cos^2 \left(\frac{\pi}{3}ight) + \cos^2 \left(\frac{\pi}{4}ight) + \cos^2 	heta = 1$$\left(\frac{1}{2}ight)^2 + \left(\frac{1}{\sqrt{2}}ight)^2 + \cos^2 	heta = 1$$\frac{1}{4} + \frac{1}{2} + \cos^2 	heta = 1$$\frac{3}{4} + \cos^2 	heta = 1$$\cos^2 	heta = 1 - \frac{3}{4} = \frac{1}{4}$$\cos 	heta = \pm \frac{1}{2}$Since $	heta \in (0, \pi)$,we have two possible values:If $\cos 	heta = \frac{1}{2}$,then $	heta = \frac{\pi}{3}$.If $\cos 	heta = -\frac{1}{2}$,then $	heta = \frac{2\pi}{3}$.Comparing with the given options,$\frac{2\pi}{3}$ is the correct value.

$A$ unit vector $\hat{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}$,$\frac{\pi}{4}$ with $\hat{j}$,and $\theta \in (0, \pi)$ with $\hat{k}$. Then a value of $\theta$ is:

Similar Questions

If the direction ratios of two lines are given by $a+2b+c=0$ and $11bc+6ca-14ab=0$, then the angle between these lines is

Show that the three lines with direction cosines $\frac{12}{13}, \frac{-3}{13}, \frac{-4}{13} ; \frac{4}{13}, \frac{12}{13}, \frac{3}{13} ; \frac{3}{13}, \frac{-4}{13}, \frac{12}{13}$ are mutually perpendicular.

The angle between a diagonal of a cube and one of its edges is ...................

If the direction cosines of two lines are given by $l+3m+5n=0$ and $5lm-2mn+6ln=0$, then the angle between the lines is

$A$ vector $\overline{OP}$ makes angles of $45^{\circ}$ with $OX$ and $60^{\circ}$ with $OY$. Find the angle that $\overline{OP}$ makes with $OZ$ (in degrees).

Vedclass Test Series

Exam Paper Generator

Online Exam Module