Scalar (Dot) Product

A vector $\vec{A}$ of magnitude $7\sqrt{2}$ unit, another vector $\vec{B}$ of magnitude $14$ unit are inclined at an angle of $60^\circ$. The magnitude of the scalar product $(\vec{A} \cdot \vec{B})$ is $\left[\cos(60)^\circ = \frac{1}{2}\right]$

In a vector $\vec{p}$, 'x' and 'y' components have magnitudes 2 units and 5 units respectively. A vector $\vec{q}$, along negative Y axis has magnitude of 3 units. Then $\vec{p} \cdot \vec{q}$ is

The vector $\vec{P} = a\hat{i} + a\hat{j} + 3\hat{k}$ and $\vec{Q} = a\hat{i} - 2\hat{j} - \hat{k}$ are perpendicular to each other. Then positive value of '$a$' is

If $\vec{A} = 5\hat{i} + 2\hat{j}$ and $\vec{B} = a\hat{i} + 3\hat{j}$ are perpendicular to each other, the value of 'a' is

If two vectors $\vec{A} = 4\hat{i} + n\hat{j} + 2\hat{k}$ and $\vec{B} = 2\hat{i} + 2\hat{j} - \hat{k}$ are mutually perpendicular to each other then value of 'n' is

The vector sum of the two forces $\vec{A}$ and $\vec{B}$ is perpendicular to their vector difference. Hence forces $\vec{A}$ and $\vec{B}$ are

The vectors $(\vec{A} + \vec{B})$ and $(\vec{A} - \vec{B})$ are perpendicular to each other. This is possible under the condition

Vectors $a\hat{i} + b\hat{j} + \hat{k}$ and $2\hat{i} - 3\hat{j} + 4\hat{k}$ are perpendicular to each other when $3a + 2b = 7$, the ratio of a to b is $x/2$. The value of x is

Calculate the values of $a$ and $b$ if vectors $a\hat{i} + b\hat{j} = \hat{n}$ and $(\hat{i} + \hat{j})$ are perpendicular to each other

Given $\vec{A} = (2\hat{i} - 3\hat{j} + \hat{k}), \quad \vec{B} = (3\hat{i} + \hat{j} - 2\hat{k})$ and $\vec{C} = (3\hat{i} + 2\hat{j} + \hat{k})$, then $(\vec{A} + \vec{B}) \cdot \vec{C}$ will be

Given $\vec{A} = (2\hat{i} - 3\hat{j} + \hat{k}), \quad \vec{B} = (3\hat{i} + \hat{j} - 2\hat{k})$ and $\vec{C} = (3\hat{i} + 2\hat{j} + \hat{k})$, then $(\vec{A} + \vec{B}) \cdot \vec{C}$ will be

The vector sum of two forces $\vec{A}$ and $\vec{B}$ is perpendicular to their vector difference. Hence forces $\vec{A}$ and $\vec{B}$ are

Two vectors $a\hat{i} + b\hat{j} + \hat{k}$ and $2\hat{i} - 3\hat{j} + 4\hat{k}$ are perpendicular to each other. When $3a + 2b = 7$, the ratio of $a$ to $b$ is $\frac{x}{2}$. The value of $x$ is

If $\vec{P} = b\hat{i} + 6\hat{j} + \hat{k}$ and $\vec{Q} = \hat{i} - a\hat{j} + 4\hat{k}$ are perpendicular to each other, also $3b - a = 5$. The value of $a$ and $b$ is

Two particles are located at equal distance from origin. The position vectors of those are represented by $\vec{A} = 2\hat{i} + 3n\hat{j} + 2\hat{k}$ and $\vec{B} = 2\hat{i} - 2\hat{j} + 4p\hat{k}$, respectively. If both the vectors are at right angle to each other, the value of $n^{-1}$ is ________.

If two vectors $\vec{P} = \hat{i} + 2m\hat{j} + m\hat{k}$ and $\vec{Q} = 4\hat{i} - 2\hat{j} + m\hat{k}$ are perpendicular to each other. Then, the value of m will be:

Vectors $a\hat{i} + b\hat{j} + \hat{k}$ and $2\hat{i} - 3\hat{j} + 4\hat{k}$ are perpendicular to each other when $3a + 2b = 7$, the ratio of $a$ to $b$ is $\frac{x}{2}$. The value of $x$ is ____________.

If the projection of $2\hat{i} + 4\hat{j} - 2\hat{k}$ on $\hat{i} + 2\hat{j} + \alpha\hat{k}$ is zero. Then, the value of $\alpha$ will be ___________.

If $\vec{A} = (2\hat{i} + 3\hat{j} - \hat{k})\text{ m}$ and $\vec{B} = (\hat{i} + 2\hat{j} + 2\hat{k})\text{ m}$. The magnitude of component of vector $\vec{A}$ along vector $\vec{B}$ will be ____________ $\text{m}$.

What will be the projection of vector $\vec{A} = \hat{i} + \hat{j} + \hat{k}$ on vector $\vec{B} = \hat{i} + \hat{j}$?

The vectors $(\mathbf{A} + \mathbf{B})$ and $(\mathbf{A} - \mathbf{B})$ are at right angles to each other. This is possible under the condition

$\mathbf{P}$ and $\mathbf{Q}$ are two non-zero vectors inclined to each other at an angle ' $\theta$ '. ' $p$ ' and ' $q$ ' are unit vectors along $\mathbf{P}$ and $\mathbf{Q}$ respectively. The component of $\mathbf{Q}$ in the direction of $\mathbf{P}$ will be

Let $\left|\vec{A}_1\right| = 3, \ \left|\vec{A}_2\right| = 5$ and $\left|\vec{A}_1 + \vec{A}_2\right| = 5$. The value of $\left(2\vec{A}_1 + 3\vec{A}_2\right) \cdot \left(3\vec{A}_1 - 2\vec{A}_2\right)$ is :-

Let $\vec{A} = (\hat{i} + \hat{j})$ and, $\vec{B} = (2\hat{i} - \hat{j}).$ The magnitude of a coplanar vector $\vec{C}$such that $\vec{A} \cdot \vec{C} = \vec{B} \cdot \vec{C} = \vec{A} \cdot \vec{B},$ is given by: