If $\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}$,the magnitude of $[(\vec{a} + 3\vec{b}) \cdot (2\vec{a} - \vec{b})]$ is

Define the scalar product and obtain the magnitude of a vector from it. Mention the direction of scalar product.

Explain the meaning of multiplication of vectors by real numbers with an example.

The angle between the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is $\theta$. The value of the triple product $\overrightarrow{A} \cdot (\overrightarrow{B} \times \overrightarrow{A})$ is

$\overrightarrow A = 2\hat i + 4\hat j + 4\hat k$ and $\overrightarrow B = 4\hat i + 2\hat j - 4\hat k$ are two vectors. The angle between them will be ........ $^o$.

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