The value of $\lambda$, for which the two vectors $2\hat{i} - \hat{j} + 2\hat{k}$ and $3\vec{i} + \lambda\vec{j} + \hat{k}$ are perpendicular, is:

If $\vec a = 4\hat i+5\hat j-\hat k, \vec b=\hat i-4\hat j+5\hat k$ and $\vec c=3\hat i+\hat j-\hat k$. If a vector $\vec d$ is perpendicular to both $\vec a$ and $\vec b$ and $\vec c.\vec d=21$, then unit vector along $\vec d$ is

Let $\vec{a} = 2\hat{i} + 3\hat{j} - 4\hat{k}$ and $\vec{b} = 3\hat{i} - 5\hat{j} + 6\hat{k}$. Let $\vec{c}$ be a vector such that $\vec{c} \times \vec{a} = \vec{b} \times \vec{c}$ and $\vec{c}\cdot(2\vec{a}-3\vec{b}) = 238\sqrt{2}$ then $|\vec{c}|^2$ is equal to :

If $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along the co-ordinate axes OX, OY and OZ respectively, then

(A) $\hat{i} \times \hat{j} = \hat{k}$

(B) $\hat{k} \times \hat{i} = -\hat{j}$

(C) $\hat{j} \cdot \hat{j} = 1$

(D) $\hat{j} \cdot \hat{k} = 0$

Choose the correct answer from the options given below:

If $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} - 3\hat{j} - 5\hat{k}$ then $(\vec{a}+\vec{b}).(\vec{a}-\vec{b})$ is equal to:

A vector $\vec{a}$ of magnitude $3\sqrt{2}$ making an angle of $\frac{\pi}{3}$ with $\hat{i}$, $\frac{\pi}{4}$ with $\hat{j}$ and an actue angle $\theta$ with $\hat{k}$, is

If $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ & $\vec{b} = \hat{i} - 3\hat{j} + 5\hat{k}$ the angle between $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ is :