If the projection of $2 \hat{i} + 4 \hat{j} - 2 \hat{k}$ on $\hat{i} + 2 \hat{j} + \alpha \hat{k}$ is zero,then the value of $\alpha$ will be.

If $|\vec{a}| = \sqrt{26}$,$|\vec{b}| = 7$,and $|\vec{a} \times \vec{b}| = 35$,find $\vec{a} \cdot \vec{b}$.

In a right-handed Cartesian coordinate system (often referred to as a clockwise system when viewed from the positive axis),which of the following vector relations is correct?

The resultant of two vectors having magnitudes $2$ and $3$ is $1$. What is their cross product?

The area of the parallelogram whose sides are represented by the vectors $\hat j + 3\hat k$ and $\hat i + 2\hat j - \hat k$ is

If two vectors $A$ and $B$ are mutually perpendicular,then the component of $A-B$ along the direction of $A+B$ is