What is the length of projection of the vector $\rm \hat{i}+2 \hat{j}+3 \hat{k}$ on the vector $\rm2 \hat{i}+3 \hat{j}-2 \hat{k}$ ?

If $\rm (\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=144$ and $\rm|\vec{b}|=4 $, then what is the value of $\rm|\vec{a}|$ ?

If θ is the angle between vectors $\vec{a}$ and $ \vec{b}$ such that $\vec{a} \cdot \vec{b} \geq 0$, then which one of the following is correct?

The vectors $\rm 60 \hat{i}+3 \hat{j}, 40 \hat{i}-8 \hat{j}$ and $\rm \beta \hat{i}-52 \hat{j}$ are collinear if :