For a unit vector $\vec{a}$,if $(\vec{x}-\vec{a}) \cdot (\vec{x}+\vec{a}) = 15$,then find $|\vec{x}|$.

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$. Then $|\vec{a}+2\vec{b}|^{2}+|\vec{a}+2\vec{c}|^{2}$ is equal to

If $A(3,4,5), B(4,6,3), C(-1,2,4)$ and $D(1,0,5)$ are such that the angle between the lines $DC$ and $AB$ is $\theta$,then $\cos \theta$ is equal to

The angle between the vectors $(2i + 6j + 3k)$ and $(12i - 4j + 3k)$ is

If $\vec{a}=2 \hat{i}+\hat{j}-\hat{k}$,$\vec{b}=\hat{i}-\hat{j}+3 \hat{k}$,$\vec{x}=\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right) \vec{b}$,$\vec{y}=\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}\right) \vec{a}$ and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$,then $x^2+y^2=$

What is the projection vector of the vector $\vec{a} = (1, 1, 1)$ onto the vector $\vec{b} = (2, 2, 1)$?