Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Vector Algebra
0 votes

If a unit vector \(\overrightarrow a\) makes angles \( \frac{\large \pi}{\large 3} \) with \( \hat i, \frac{\large \pi}{\large 3} \) with \( \hat j\) and an acute angle \(θ\) with \( \hat k\) , then find \( θ \) and hence, the components of \( \overrightarrow a \) .

Can you answer this question?

1 Answer

0 votes
  • Sum of the squares of the direction cosines is $\cos^2\theta_1+\cos^2\theta_2+\cos^2\theta_3=1$
Step 1:
It is given that $\overrightarrow a$ makes an angle $\large\frac{\pi}{3}$ with $\hat i$ and $\large\frac{\pi}{4}$ with $\hat j$ and acute angle with $\hat k$
Let us find the acute angle $\theta$ made with $\hat k$
We know sum of the squares of the direction cosines of angles made with $\hat i,\hat j$ and $\hat k$ is 1.
(i.e) $\cos^2\theta_1+\cos^2\theta_2+\cos^2\theta_3$=1.
Here $\cos\theta_1=\cos\large\frac{\pi}{3}=\large\frac{1}{2}$
$\quad\cos\theta_2=\cos\large\frac{\pi}{4}=\large\frac{1}{\sqrt 2}$
Let the third angle be $\theta_3$
Therefore $\big(\large\frac{1}{2}\big)^2+\big(\large\frac{1}{\sqrt 2}\big)^2+$$\cos^2\theta_3=1$
Step 2:
$\cos^2\theta_3=\large\frac{1}{4}\Rightarrow $$\cos\theta_3=\pm\large\frac{1}{2}$
Step 3:
But it is said that $\theta_3$ is an acute angle.
Therefore $\cos\theta_3=\large\frac{1}{2}$
$\Rightarrow \theta_3=\cos^{-1}\big(\large\frac{1}{2}\big)$
Step 4:
Hence the angle made with $\hat k$ is $\large\frac{\pi}{3}$
The components of $\overrightarrow a$ are $\large\frac{1}{2},\frac{1}{\sqrt 2}$ and $\large\frac{1}{2}$
answered May 22, 2013 by sreemathi.v

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App