NIMCET 2025 — Mathematics PYQ

Step 1: Calculate the dot product  $\vec{a}\cdot \vec{b}$  

$\vec{a}\cdot \vec{b}=(2)(-2)+(3)(1)+(1)(2)$  

$\vec{a}\cdot \vec{b}=-4+3+2$  

$\vec{a}\cdot \vec{b}=1$  

Step 2: Calculate the magnitude of  $\vec{b}$  

$|\vec{b}|=\sqrt{(-2{)}^2+(1{)}^2+(2{)}^2}$  

$|\vec{b}|=\sqrt{4+1+4}$  

$|\vec{b}|=\sqrt{9}=3$  

Step 3: Calculate the scalar projection 

projection of $\vec{b}$ on $\vec{a}$ = $\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}$  

 projection of $\vec{b}$ on $\vec{a}$= $\frac{1}{3}$  

Solution: The length of the projection of $\vec{a}$ on $\vec{b}$ is $\frac{1}{3}$.