Question

Determine the angle which the vector $A=5i+6j+5k$ makes with $x,y,z$ axis.

Answer 1

Unit vector along X axis =$\hat{i}$

Unit vector along Y axis =$\hat{j}$

Unit vector along Z axis =$\hat{k}$

We use dot product to find angle

$\overrightarrow{a}.\overrightarrow{b}=ab\cos \theta$

Now with X axis:

$\overrightarrow{A}.\hat{i}=A\times 1\times \cos \theta$

$(5\hat{i}+0\hat{j}+5\hat{k}).\hat{i}=5\sqrt{2}\cos \theta$

$5=5\sqrt{2}\cos \theta$

$\therefore \cos \theta =\frac{1}{\sqrt{2}}$

$\theta =\frac{\pi }{4}$

Now with Y axis:

$\overrightarrow{A}.\hat{j}=A\times 1\times \cos \theta$

$(5\hat{i}+0\hat{j}+5\hat{k}).\hat{j}=5\sqrt{2}\cos \theta$

$0=5\sqrt{2}\cos \theta$

$\therefore \cos \theta =0$

$\theta =\frac{\pi }{2}$

Now with Z axis:

$\overrightarrow{A}.\hat{k}=A\times 1\times \cos \theta$

$(5\hat{i}+0\hat{j}+5\hat{k}).\hat{k}=5\sqrt{2}\cos \theta$

$5=5\sqrt{2}\cos \theta$

$\therefore \cos \theta =\frac{1}{\sqrt{2}}$

$\theta =\frac{\pi }{4}$