If a vector has direction angles 45° and 60°, find the third dire… - Vidyadip

Question

If a vector has direction angles 45° and 60°, find the third direction angle.

✓

Answer

Let $\alpha, \beta, \gamma$ be the angles made by the vector with positive directions of $X, Y, Z$ axes respectively.
$\therefore \alpha=45^{\circ}, \beta=60^{\circ}$
We know that,
$ \because \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1$
$\therefore \cos ^2 45^{\circ}+\cos ^2 60^{\circ}+\cos ^2 r=1 $
$\therefore\left(\frac{1}{\sqrt{2}}\right)^2+\left(\frac{1}{2}\right)^2+\cos ^2 \gamma=1$
$\therefore \frac{1}{2}+\frac{1}{4}+\cos ^2 \gamma=1$
$ \therefore \cos ^2 \gamma=1-\frac{1}{2}-\frac{1}{4}$
$\therefore \cos ^2 \gamma=\frac{1}{4} $
$\therefore \cos ^2 \gamma=\frac{1}{4}$
$\therefore \cos \gamma= \pm \frac{1}{2}$
$\therefore \cos \gamma=\frac{1}{2}$ or $\cos \gamma=-\frac{1}{2}$
$\therefore \cos \gamma=\frac{\pi}{3}$ or $\cos \gamma=-\frac{\pi}{3}$
$\therefore \cos \left(\pi-\frac{\pi}{3}\right)=\cos \frac{2 \pi}{3}$
$\therefore \gamma=\frac{\pi}{3}$ or $\gamma=\frac{2 \pi}{3}$
Hence, the third direction angle is $\frac{\pi}{3}$ or $\frac{2 \pi}{3}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(2 Marks)" from Question Bank [2022]→Question Bank [2022] — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

When three dice are thrown, write the probability of getting 4 or 5 on each of the dice simultaneously.

If $\bar{a} \cdot \bar{b}=\sqrt{3}$ and $\bar{a} \times \bar{b}=2 \hat{i}+\hat{j}+2 \hat{k}$, find the angle between $\bar{a}$ and $\bar{b}$.

The probability distribution of $X$ is as follows:

$X=x$	$0$	$1$	$2$	$3$	$4$
$P(X=x)$	$0.1$	$K$	$2K$	$2K$	$K$

Find $(i) \ K$ $(ii) \ P(X < 2)$

For the set A = {1, 2, 3}, define a relation R on the set A as follows:
R = {(1, 1), (2, 2), (3, 3), (1, 3)}
Write the ordered pairs to be added to R to make the smallest equivalence relation.

Find the angle betwwen two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ if$|\vec{\text{a}}|=3,\big|\vec{\text{b}}\big|=3$ and $\vec{\text{a}}.\vec{\text{b}}=1$

For the following differntial equations verify that the accompanying function is a solution:

Differential equation	Function
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}$	$\text{y}=\text{ax}$

Prove that : (i) [latex]\bar{a} \bar{b}+\bar{c} \bar{a}+\bar{b}+\bar{c}[/latex] = 0

Integrate the following functions w.r.t. x:

$\sqrt{2 x^2+3 x+4}$

Without using truth table prove that:

~ (p ∨ q) ∨ (~ p ∧ q) ≡ ~ p

Solve graphically :y ≤ 0