If $\vec{a}$ and $\vec{b}$ are two unit vectors inclined at an angle $\frac{\pi}{3}$,then the value of $|\vec{a}+\vec{b}|$ is

For non-zero vectors $a$ and $b$, if $|a+b| < |a-b|$, then $a$ and $b$ are

Let $\vec{a}=2 \hat{i}+5 \hat{j}-\hat{k}$,$\vec{b}=2 \hat{i}-2 \hat{j}+2 \hat{k}$ and $\vec{c}$ be three vectors such that $(\vec{c}+\hat{i}) \times (\vec{a}+\vec{b}+\hat{i}) = \vec{a} \times (\vec{c}+\hat{i})$ and $\vec{a} \cdot \vec{c} = -29$. Then $\vec{c} \cdot (-2 \hat{i}+\hat{j}+\hat{k})$ is equal to:

The orthogonal projection of vector $\vec{a}$ on vector $\vec{b}$ is:

Find the angle between the vectors $\hat{i}-2 \hat{j}+3 \hat{k}$ and $3 \hat{i}-2 \hat{j}+\hat{k}$.

Let $\vec{a} = 6 \hat{i} - 3 \hat{j} - 6 \hat{k}$ and $\vec{d} = \hat{i} + \hat{j} + \hat{k}$. Suppose that $\vec{a} = \vec{b} + \vec{c}$,where $\vec{b}$ is parallel to $\vec{d}$ and $\vec{c}$ is perpendicular to $\vec{d}$. Then $\vec{c}$ is