The angle between the pair of lines with direction ratios $(1, 1, 2)$ and $(\sqrt{3} - 1, -\sqrt{3} - 1, 4)$ is ......... $^o$.

Let $a, b, c \in \mathbb{R}$ be such that $a^{2} + b^{2} + c^{2} = 1$. If $a \cos \theta = b \cos \left(\theta + \frac{2\pi}{3}\right) = c \cos \left(\theta + \frac{4\pi}{3}\right)$ where $\theta = \frac{\pi}{9}$,then the angle between the vectors $\vec{p} = a \hat{i} + b \hat{j} + c \hat{k}$ and $\vec{q} = b \hat{i} + c \hat{j} + a \hat{k}$ is:

Let $a = 2\hat{i} - \hat{j} + 2\hat{k}$ and $b = 3\hat{i} - 2\hat{j} - 5\hat{k}$ be two vectors. Then the projection vector of $b$ on a vector perpendicular to $a$ is

If $|\vec{a}|=1, |\vec{b}|=2, |\vec{a}-\vec{b}|^2+|\vec{a}+2\vec{b}|^2=20$, then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is:

Let $a=\hat{i}+2 \hat{j}-2 \hat{k}$ and $b=2 \hat{i}-\hat{j}-2 \hat{k}$. If the orthogonal projection vector of $a$ on $b$ is $x$ and the orthogonal projection vector of $b$ on $a$ is $y$, then $|x-y|=$

If $|a| = 3$ and $|b| = 4$,then a value of $\lambda$ for which $a + \lambda b$ is perpendicular to $a - \lambda b$ is