The two vectors have magnitudes $3$ and $5$. If angle between them is $60^o$, then the dot product of two vectors will be

What will be the projection of vector $A=\hat{i}+\hat{j}+\hat{k}$ on vector $\vec{B}=\hat{i}+\hat{j}$.

Let $\left| {{{\vec A}_1}} \right| = 3,\,\left| {\vec A_2} \right| = 5$, and $\left| {{{\vec A}_1} + {{\vec A}_2}} \right| = 5$. The value of $\left( {2{{\vec A}_1} + 3{{\vec A}_2}} \right)\cdot \left( {3{{\vec A}_1} - 2{{\vec A}_2}} \right)$ is

The value of $(\overrightarrow A + \overrightarrow B )\, \times (\overrightarrow A - \overrightarrow B )$ is

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

Vectors $a \hat{i}+b \hat{j}+\hat{k}$ and $2 \hat{i}-3 \hat{j}+4 \hat{k}$ are perpendicular to each other when $3 a+2 b=7$, the ratio of a to $b$ is $\frac{x}{2}$. The value of $x$ is $..............$