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

If $a \times r = b + \lambda a$ and $a \cdot r = 3,$ where $a = 2i + j - k$ and $b = -i - 2j + k,$ then $r$ and $\lambda$ are equal to

If the vector $a = 3\hat{j} + 4\hat{k}$ is the sum of two vectors $a_1$ and $a_2$, where vector $a_1$ is parallel to $b = \hat{i} + \hat{j}$ and vector $a_2$ is perpendicular to $b$, then $a_1 =$

Let $a = i + 2j + k$,$b = i - j + k$,$c = i + j - k$. $A$ vector in the plane of $a$ and $b$ has projection $\frac{1}{\sqrt{3}}$ on $c$. Then,one such vector is

If $\vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$ are orthogonal,then the value of $\lambda$ 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: