Vector Algebra

The velocity of a particle is given as $\vec{v} = -x \hat{i} + 2y \hat{j} - z \hat{k}$ m/s. The magnitude of acceleration at point (1, 2, 4) is ________ m/s².

The angle between vector $\vec{Q}$ and the resultant of $(2 \vec{Q}+2 \vec{P})$ and $(2 \vec{Q}-2 \vec{P})$ is :

If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $R$ are inclined at angle $\theta$, then

When vector $\vec{A}=2 \hat{i}+3 \hat{j}+2 \hat{k}$ is subtracted from vector $\overrightarrow{\mathrm{B}}$, it gives a vector equal to $2 \hat{j}$. Then the magnitude of vector $\overrightarrow{\mathrm{B}}$ will be :

Two forces having magnitude $A$ and $\frac{A}{2}$ are perpendicular to each other. The magnitude of their resultant is:

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

Two vectors $\overrightarrow A$ and $\overrightarrow B$ have equal magnitudes. If magnitude of $\overrightarrow A$ + $\overrightarrow B$ is equal to two times the magnitude of $\overrightarrow A$ $-$ $\overrightarrow B$, then the angle between $\overrightarrow A$ and $\overrightarrow B$ will be :

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

Which of the following relations is true for two unit vector $\widehat A$ and $\widehat B$ making an angle $\theta$ to each other?

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

The resultant of two vectors $\vec{A}$ and $\vec{B}$ is perpendicular to $\vec{A}$ and its magnitude is half that of $\vec{B}$. The angle between vectors $\vec{A}$ and $\vec{B}$ is _________$^\circ$.

If $\vec{a}$ and $\vec{b}$ makes an angle $\cos ^{-1}\left(\frac{5}{9}\right)$ with each other, then $|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}|$ for $|\vec{a}|=n|\vec{b}|$ The integer value of $\mathrm{n}$ is _________.

Three vectors $\overrightarrow{\mathrm{OP}}, \overrightarrow{\mathrm{OQ}}$ and $\overrightarrow{\mathrm{OR}}$ each of magnitude $\mathrm{A}$ are acting as shown in figure. The resultant of the three vectors is $\mathrm{A} \sqrt{x}$. The value of $x$ is _________.

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 ________.

A vector has magnitude same as that of $\vec{A}=3 \hat{i}+4 \hat{j}$ and is parallel to $\vec{B}=4 \hat{i}+3 \hat{j}$. The $x$ and $y$ components of this vector in first quadrant are $x$ and 3 respectively where $x=$ _________.

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

Vectors $a\widehat i + b\widehat j + \widehat k$ and $2\widehat i - 3\widehat j + 4\widehat k$ are perpendicular to each other when $3a + 2b = 7$, the ratio of $a$ to $b$ is ${x \over 2}$. 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 $\vec{A}=(2 \hat{i}+3 \hat{j}-\hat{k})\, \mathrm{m}$ and $\vec{B}=(\hat{i}+2 \hat{j}+2 \hat{k}) \,\mathrm{m}$. The magnitude of component of vector $\vec{A}$ along vector $\vec{B}$ will be ____________ $\mathrm{m}$.