Vector (Cross) Product

The area of parallelogram formed by vectors $\vec{P} = 2\hat{i} - \hat{j} + 5\hat{k}$ and $\vec{Q} = 3\hat{i} - 2\hat{j} + 4\hat{k}$ is

If $|\vec{A} \times \vec{B}| = \sqrt{3}(\vec{A} \cdot \vec{B})$ then the value of $|\vec{A} + \vec{B}|$ is ($\tan 60^\circ = \sqrt{3} \quad \cos 60^\circ = \frac{1}{2}$)

For two vectors $\vec{P}$ and $\vec{Q} \quad \vec{P} \cdot \vec{Q} = |\vec{P} \times \vec{Q}|$

The magnitude of $\vec{R} = \vec{P} + \vec{Q}$ is ($\cos 45^\circ = \frac{1}{\sqrt{2}}$)?

If $|\vec{a}| = \sqrt{26}, \ |\vec{b}| = 7$

$|\vec{a} \times \vec{b}| = 35$, find $\vec{a} \cdot \vec{b}$

Vector $\vec{A}$ of magnitude $5\sqrt{3}$ units, another vector $\vec{B}$ of magnitude of 10 units are inclined to each other at an angle of $30^\circ$. The magnitude of vector product of the two vectors is $\left[\sin 30^\circ = \frac{1}{2}\right]$

If $\vec{A} = \hat{i} + \hat{j} + 3\hat{k}, \ \vec{B} = -\hat{i} + \hat{j} + 4\hat{k}$ and $\vec{C} = 2\hat{i} - 2\hat{j} - 8\hat{k}$, then the angle between the vectors $\vec{P} = \vec{A} + \vec{B} + \vec{C}$ and $\vec{Q} = (\vec{A} \times \vec{B})$ is (in degree)

For three vectors $\vec{A} = (-x\hat{i} - 6\hat{j} - 2\hat{k}), \ \vec{B} = (-\hat{i} + 4\hat{j} + 3\hat{k})$ and $\vec{C} = (-8\hat{i} - \hat{j} + 3\hat{k})$, if $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$, then value of $x$ is ________.

If $\vec{P} = 3\hat{i} + \sqrt{3}\hat{j} + 2\hat{k}$ and $\vec{Q} = 4\hat{i} + \sqrt{3}\hat{j} + 2.5\hat{k}$ then, the unit vector in the direction of $\vec{P} \times \vec{Q}$is $\frac{1}{x}\left(\sqrt{3}\hat{i} + \hat{j} - 2\sqrt{3}\hat{k}\right)$. The value of $x$ is _________.

If $\vec{F} = 2\hat{i} + \hat{j} - \hat{k}$ and$\vec{r} = 3\hat{i} + 2\hat{j} - 2\hat{k}$, then the scalar and vector products of $\vec{F}$and$\vec{r}$have the magnitudes respectively as

$\vec{A}$is a vector quantity such that $|\vec{A}|$ = non-zero constant. Which of the following expression is true for $\vec{A}$?

If $\vec{P} \times \vec{Q} = \vec{Q} \times \vec{P}$, the angle between $\vec{P}$and $\vec{Q}$ is $\theta(0^\circ$ < $\theta$ < 360$^\circ$). The value of '$\theta$' will be ___________$^\circ$.

If $\vec{A}$and $\vec{B}$ are two vectors satisfying the relation $\vec{A} \quad \vec{B}$. = . Then the value of will be:

Three particles P, Q and R are moving along the vectors $\vec{A} = \hat{i} + \hat{j} \quad \vec{B} = \hat{j} + \hat{k}$ and $\vec{C} = -\hat{i} + \hat{j}$ respectively. They strike on a point and start to move in different directions. Now particle P is moving normal to the plane which contains vector $\vec{A}$and $\vec{B}$. Similarly particle Q is moving normal to the plane which contains vector $\vec{A}$and $\vec{C}$. The angle between the direction of motion of P and Q is $\cos^{-1}\left(\frac{1}{\sqrt{x}}\right)$. Then the value of x is _______________.

What is the angle between resultant of $A + B$ and $\mathbf{A} \times \mathbf{B}$.

If $\vec{A} \times \vec{B} = \vec{B} \times \vec{A}$, then the angle between A and B is

What is the linear velocity if angular velocity vector $\vec{\omega} = 3\hat{i} - 4\hat{j} + \hat{k}$ and position vector $\vec{r} = 5\hat{i} - 6\hat{j} + 6\hat{k}$?