The projection of vector $\vec{a}$ along vector $\vec{b}$ 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:

The constant value $(\lambda + \mu)$ for which the lines $\vec{r} = (2\hat{i} + \hat{j} + \hat{k}) + \lambda(\hat{i} - 2\hat{j})$ and $\vec{r} = (\hat{i} + \hat{j} - 3\hat{k}) + \mu(\hat{j} + 2\hat{k})$ intersect each other is equal to (where $\lambda$ and $\mu$ are parameters).

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\sqrt{3} \vec{c}=\overrightarrow{0}$,then the angle between $\vec{a}$ and $\vec{b}$ is

Let $\vec{a} \times \vec{b} = 7 \hat{i} - 5 \hat{j} - 4 \hat{k}$ and $\vec{a} = \hat{i} + 3 \hat{j} - 2 \hat{k}$. If the length of the projection of $\vec{b}$ on $\vec{a}$ is $\frac{8}{\sqrt{14}}$,then $|\vec{b}| = $

Assertion $(A)$: $a, b, c, d$ are position vectors of $4$ points such that $2a - 3b + 7c - 6d = 0 \Rightarrow a, b, c, d$ are coplanar.
Reason $(R)$: Vector equation of the plane passing through three points whose position vectors are $a, b, c$ is $r = (1 - x - y)a + xb + yc$.
Which of the following is true?