Dot Product

$ \vec{a} = x\hat{i} - 3\hat{j} - \hat{k},\quad \vec{b} = 2x\hat{i} + x\hat{j} - \hat{k} $ Angle between $ \vec{a} $ and $ \vec{b} $ is acute and angle between $ \vec{b} $ and $ +y $ axis lies in $ \left(\dfrac{\pi}{2}, \pi\right) $ Find $x$.

Force $3\hat{i}+2\hat{j}+5\hat{k}$ and $2\hat{i}+\hat{j}-3\hat{k}$ are acting on a particle and displace it from the point $2\hat{i}-\hat{j}-3\hat{k}$ to $4\hat{i}-3\hat{j}+7\hat{k}$ the point then the work done by the force is

Let $\vec{a}=\hat{j}-\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ . Then the vector $\vec{b}$ satisfying $(\vec{a} \times \vec{b})+ \vec{c} =0$ and $\vec{a} . \vec{b}=3$ is

The sum of two vectors $\vec{a}$ and $\vec{b}$ is a vector $\vec{c}$ such that $|\vec{a}|=|\vec{b}|=|\vec{c}|=2$. Then, the magnitude of $\vec{a}-\vec{b}$ is equal to:

Constant forces $\vec{P}= 2\hat{i} - 5\hat{j} + 6\hat{k} $ and $\vec{Q}= -\hat{i} + 2\hat{j}- \hat{k}$  act on a particle. The work done when the particle is displaced from A whose position vector is $4\hat{i} - 3\hat{j} - 2\hat{k} $, to B whose position vector is $6\hat{i} + \hat{j} - 3k\hat{k}$ , is:

If $\vec{a}$ and $\vec{b}$ are vectors such that $|\vec{a}|=13$, $|\vec{b}|=5$ and $\vec{a} . \vec{b} =60$then the value of $|\vec{a} \times \vec{b}|$ is

Let $\vec{a}$ and $\vec{b}$ be two vectors, which of the following vectors are not perpendicular to each other?

Let ,  and  be three vector such that || = 2, || = 3, || = 5 and ++ = 0. The value of .+.+. is