Let $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$. If $\vec{b}$ is a vector such that $\vec{a} \cdot \vec{b} = |\vec{b}|^2$ and $|\vec{a} - \vec{b}| = \sqrt{7}$,then find $|\vec{b}|$.

Let $\vec{\alpha}=4 \hat{i}+3 \hat{j}+5 \hat{k}$ and $\vec{\beta}=\hat{i}+2 \hat{j}-4 \hat{k}$. Let $\vec{\beta}_1$ be parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ be perpendicular to $\vec{\alpha}$. If $\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2$,then the value of $5 \vec{\beta}_2 \cdot(\hat{i}+\hat{j}+\hat{k})$ is

If $\overline{a}=2 \hat{i}+3 \hat{j}+2 \hat{k}$,$\overline{b}=2 \hat{i}+\hat{j}-\hat{k}$ and $\overline{c}=\hat{i}+3 \hat{j}$ are such that $(\overline{a}+\lambda \overline{b})$ is perpendicular to $\overline{c}$,then the value of $\lambda$ is

The value of $b$ such that the scalar product of the vector $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ with the unit vector parallel to the sum of the vectors $\vec{u} = 2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\vec{v} = b\hat{i} + 2\hat{j} + 3\hat{k}$ is $1$, is...

If $A, B, C, D$ are the points with position vectors $\hat{i}-\hat{j}+\hat{k}, 2 \hat{i}-\hat{j}+3 \hat{k}, 2 \hat{i}-3 \hat{k}$ and $3 \hat{i}-2 \hat{j}+\hat{k}$ respectively,find the projection of $\overrightarrow{AB}$ on $\overrightarrow{CD}$.

Three forces $i + 2j - 3k$,$2i + 3j + 4k$,and $i - j + k$ are acting on a particle at the point $(0, 1, 2)$. The magnitude of the moment of the forces about the point $(1, -2, 0)$ is