Question

The axes of co-ordinates are rotated about z-axis through an angle of $\pi /4$ in anti-clockwise direction and the components of a vector are $2\sqrt{2},3\sqrt{2},4.$ Then the components of the same vector in the original system are -1, 5, 4.

• A. $a=5i-j-4k$
• B. $a=-5i+j+4k$
• C. $a=i-5j-4k$
• D. $a=-i+5j+4k$

Answer 1

The correct option is D $a=-i+5j+4k$

If i', j', k' be new unit vectors along the co-ordinate axes, then

$a=2\sqrt{2i^{\prime }}+3\sqrt{2j^{\prime }}+4k^{\prime }$ ..(1)

i', j', k' are obtained by rotating by ${45}^{\circ }$ about z-axis.

$i^{\prime }=i\cos {45}^{\circ }+j\sin {45}^{\circ }=\frac{i+j}{\sqrt{2}}$

$j^{\prime }=i\cos {45}^{\circ }+j\sin {45}^{\circ }=\frac{-i+j}{\sqrt{2}}$

$k^{\prime }=k,$ Putting in (1)

$a=2\sqrt{2}\left[\frac{i+j}{\sqrt{2}}\right]+3\sqrt{2}\left[\frac{-i+j}{\sqrt{2}}\right]+4k$

$a=(2-3)i+(2+3)j+4k$

$a=-i+5j+4k$