If $a$ and $b$ are two unit vectors such that $a+2b$ and $5a - 4b$ are perpendicular to each other,then the angle between $a$ and $b$ is ............. $^o$

If vectors $\bar{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}$,$\bar{b}=-\hat{i}+2 \hat{j}+\hat{k}$,and $\bar{c}=-3 \hat{i}+\hat{j}+2 \hat{k}$ are such that $\bar{a}+\lambda \bar{b}$ is perpendicular to $\bar{c}$,then $\lambda=$

If $a$ and $b$ are vectors such that $|a+b| = |a-b|$,then the angle between $a$ and $b$ is (in $^{\circ}$)

Let $\vec{a} = \sqrt{7}\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{k}$. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a} + \vec{a} \times \vec{b} = \vec{0}$ and $\vec{r} \cdot \vec{a} = 0$, then $|3\vec{r}|^2$ is equal to:

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

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: