If $a$ and $b$ are unit vectors and $\theta$ is the angle between $a$ and $b$,then $\sin \frac{\theta}{2}$ is equal to

Let $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=2 \hat{j}-3 \hat{k}$. If $\vec{b}=\vec{c}-\vec{d}$,$\vec{a}$ is parallel to $\vec{c}$,and $\vec{a}$ is perpendicular to $\vec{d}$,then $\vec{c}+\vec{d}=$

Let $a, b$ and $c$ be vectors with magnitudes $3, 4$ and $5$ respectively and $a + b + c = 0$. Then the value of $a \cdot b + b \cdot c + c \cdot a$ is:

If the angle between the vectors $\vec{a} = 2\lambda^2 \hat{i} + 4\lambda \hat{j} + \hat{k}$ and $\vec{b} = 7\hat{i} - 2\hat{j} + \lambda \hat{k}$ is obtuse,then the values of $\lambda$ lie in:

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}$ is perpendicular to both $\vec{b}$ and $\vec{c}$,and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2 \pi}{3}$,then $|\vec{a}+3 \vec{b}-4 \vec{c}|^2=$

If $\bar{a}$ and $\bar{b}$ are two unit vectors such that $\bar{a}+2 \bar{b}$ and $5 \bar{a}-4 \bar{b}$ are perpendicular to each other,then the angle between $\bar{a}$ and $\bar{b}$ is