$A$ vector coplanar with vectors $i + j$ and $j + k$ and parallel to the vector $2i - 2j - 4k$ is:

Let $\vec{a}=4 \hat{i}+3 \hat{j}$ and $\vec{b}=3 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\vec{c}$ is a vector such that $\vec{c} \cdot(\vec{a} \times \vec{b})+25=0, \vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})=4$ and the projection of $\vec{c}$ on $\vec{a}$ is $1$. Then,the projection of $\vec{c}$ on $\vec{b}$ equals:

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

If $\alpha=\hat{i}-3 \hat{j}$ and $\beta=\hat{i}+2 \hat{j}-\hat{k}$,express $\beta$ in the form $\beta=\beta_1+\beta_2$,where $\beta_1$ is parallel to $\alpha$ and $\beta_2$ is perpendicular to $\alpha$. Then $\beta_1$ is given by:

For what value of $x$ is the angle between the vectors $\vec{a} = -3\hat{i} + x\hat{j} + \hat{k}$ and $\vec{b} = x\hat{i} + 2x\hat{j} + \hat{k}$ acute,and the angle between $\vec{b}$ and the $x$-axis lies between $\pi/2$ and $\pi$?

The angle between two diagonals of a cube is: