Solveeit Logo
Login

Question

Mathematics Question on Vectors

Let a⃗\vec a=i^\hat i−j^\hat j+2k^\hat k and b⃗\vec b be a vector such that
a⃗\vec a×b⃗\vec b=2i^\hat i−k^\hat k and a⃗\vec a⋅b⃗\vec b=3.
Then the projection of b⃗\vec b on the vector a⃗\vec a−b⃗\vec b is :

A

221\frac{2}{\sqrt{21}}

B

237\frac{2\sqrt3}{7}

C

2373\frac{2}{\frac{3\sqrt7}{3}}

D

23\frac{2}{3}

Answer

221\frac{2}{\sqrt{21}}

Explanation

Solution

a⃗\vec a=i^\hat i−j^\hat j+2k^\hat k
a⃗\vec a×b⃗\vec b=2i^\hat i−k^\hat k
a⃗\vec a⋅b⃗\vec b=3 |a⃗\vec a×b⃗\vec b|2+|a⃗\vec a⋅b⃗\vec b|2
=|a⃗\vec a|2⋅|b⃗\vec b|2
⇒5+9=6|b⃗\vec b|2
⇒|b|2=73\frac{7}{3}
|a⃗\vec a−b⃗\vec b|=√|a⃗\vec a|2+|b⃗\vec b|2−2a⃗\vec a⋅b⃗\vec b=73\frac{\sqrt7}{3}
projection of b⃗\vec b on a⃗\vec a−b⃗\vec b=b⃗(a⃗−b⃗)∣a⃗−b⃗∣\frac{\vec b(\vec a-\vec b)}{|\vec a-\vec b|}
=221\frac{2}{\sqrt{21}}