Algebra Questions - JEE

Let $\vec{a}=\hat{i}+\alpha\hat{j}+\beta\hat{k}$, $\alpha,\beta\in\mathbb{R}$. Let $\vec{b}$ be a vector such that the angle between $\vec{a}$ and $\vec{b}$ is $\dfrac{\pi}{4}$ and $|\vec{b}|^2=6$. If $\vec{a}\cdot\vec{b}=3\sqrt{2}$, then the value of $(\alpha^2+\beta^2)|\vec{a}\times\vec{b}|^2$ is equal to:

If O is the origin and the position vector of A is 4i + 5j, then a unit vector parallel to OA is

Find the volume of a parallelepiped with coterminous edges of lengths 8, 6, and 2.

Let a, b, c be three non-coplanar vectors and d be a non-zero vector, which is perpendicular to a + b + c. If d = $\sin x$(\a × b) + $\cos y$(b × c) + 2(c × a), then the minimum value of $x^2 + y^2$ is

If x, y are two non-zero and non-collinear vectors satisfying \[[(a-2)\alpha^2 + (b-3)\alpha + c]\mathbf{x} + [(a-2)\beta^2 + (b-3)\beta + c]\mathbf{y} + [(a-2)\gamma^2 + (b-3)\gamma + c](\mathbf{x} \times \mathbf{y}) = 0\] where $\alpha, \beta, \gamma$ are three distinct real numbers, then find the value of $a^2 + b^2 + c^2 - 4$.

Let a = $\alpha \mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$, b = $\mathbf{i} + 2\alpha \mathbf{j} - 2\mathbf{k}$ and c = $2\mathbf{i} - \alpha \mathbf{j} + \mathbf{k}$. Then the value of $\alpha$ such that $\{(\mathbf{a} \times \mathbf{b}) \times (\mathbf{b} \times \mathbf{c})\} \times (\mathbf{c} \times \mathbf{a}) = 0$, is

A vector a has components 2p and 1 with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If with respect to the new system, a has components (p+1) and 1, then find p.

Max \{|PQ|\} is

If a = 2i − 3j + k, b = −i + k, c = 2j − k, then the area (in sq units) of parallelogram with diagonals a + b and b + c will be

If D, E and F are respectively the mid-points of AB, AC and BC in $\triangle ABC$, then BE + AF is equal to

If a × (b × c) = 0, then

Let $ABCD$ be a tetrahedron in which position vectors of $A, B, C$ and $D$ are $\vec{i} + \vec{j} + \vec{k}, 2\vec{i} + 2\vec{j} + 2\vec{k}, 3\vec{i} + 2\vec{j} + \vec{k}$ and $2\vec{i} + 3\vec{j} + 2\vec{k}$. If $ABC$ be the base of tetrahedron then height of tetrahedron is:

The position vector of a point C with respect to B is $\mathbf{i} + \mathbf{j}$ and that of B with respect to A is $\mathbf{i} - \mathbf{j}$. Find the position vector of C with respect to A.

The value of $\frac{q}{p} + 2\cos\theta$ is

Let a = 2i + j − 2k, b = i + j and c be a vector such that |c − a| = 3, |(a × b) × c| = 3 and the angle between c and a × b is 30°. Then a · c is equal to

If $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are four non-coplanar unit vectors, $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular, such that $\vec{d}$ makes equal angles with all the three vectors $\vec{a}, \vec{b}, \vec{c}$, then:

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a} \cdot \vec{b} = 0 = \vec{a} \cdot \vec{c}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$. Then the value of $|\vec{a} \times \vec{b} - \vec{a} \times \vec{c}|$ is ______.

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ be such that $\vec{c} - 3\vec{a} = -\vec{b} - \vec{a}$. Then the points with position vectors as $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are:

253. Let $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors satisfying $\vec{a} = \vec{b} \times \vec{c} + 2\vec{b}$ where $|\vec{b}| = |\vec{c}| = 2$ and $|\vec{a}| \leq 4$. The sum of possible value(s) of $|2\vec{a} + \vec{b} + \vec{c}|$ is:

If the equation $x^3+bx^2+cx+1=0$ ($b<c$) has only one real root $\alpha$. Then the value of $2\tan^{-1}(\text{cosec}\,\alpha)+\tan^{-1}(2\sin\alpha\sec^2\alpha)$ is

Line $L_1$ is parallel to a vector $\vec{a} = 3\hat{i} + 2\hat{j} + 4\hat{k}$ and passes through a point $A(7,6,2)$ and the line $L_2$ is parallel to a vector $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$ and passes through a point $B(5, 3, 4)$. Now a line $L_3$ parallel to a vector $\vec{c} = 2\hat{i} + 2\hat{j} + \hat{k}$ intersects the lines $L_1$ and $L_2$ at points $C$ and $D$ respectively then $|\overrightarrow{CD}|$ is_______.

Let $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$. If $\vec{c}$ is a vector such that $\vec{a}\vec{c} = |\vec{c}||\vec{a} - \vec{a}| = 2\sqrt{2}$ and the angle between $(\vec{a} \times \vec{b})$ and $\vec{c}$ is $30°$, then the value of $|(\vec{a} \times \vec{b}) \times 2\vec{c}|$ is _______.

If $\vec{a} \times (\vec{b} \times \vec{c}) + (\vec{a}\vec{b})\vec{b} = (4 - 2\beta - \sin \alpha)\vec{b} + (\beta^2 - 1)\vec{c}$ and $\vec{c} \cdot \vec{c} = \vec{a} \cdot \vec{c}$ where $\vec{b}$ and $\vec{c}$ are non-collinear and $\alpha, \beta$ are scalars then $\beta = _______.

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = x_1\hat{i} + x_2\hat{j} + x_3\hat{k}$, where $x_1, x_2, x_3 \in \{-3, -2, -1, 0, 1, 2\}$. Number of possible vectors $\vec{b}$ such that $\vec{a}$ and $\vec{b}$ are mutually perpendicular, is $p$ then $\frac{p}{5} = _______.

If in a triangle $ABC$, $\overrightarrow{BC} = \frac{\vec{u}}{|\vec{u}|\vec{v}|}$ and $\overrightarrow{AC} = \frac{2\vec{u}}{|\vec{u}|}$ where $|\vec{u}||\vec{v}|$, then $1 + \cos 2A + \cos 2B + \cos 2C = _______.

Let $\vec{u}$ and $\vec{v}$ are unit vectors and $\vec{w}$ is a vector such that $\vec{u} \times \vec{v} + \vec{u} = \vec{w}$ and $\vec{w} \times \vec{u} = \vec{v}$. Then the value of $[\vec{u}\vec{v}\vec{w}] = _______.

Let $A, B, C$ be points with position vectors $\vec{r_1} = 2\hat{i} - \hat{j} + \hat{k}, \vec{r_2} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{r_3} = 3\hat{i} + \hat{j} + 2\hat{k}$ relative to the origin 'O'. Find the shortest distance between point $B$ and plane $OAC$.

Volume of tetrahedron whose vertices are the points with position vectors $\hat{i} - 6\hat{j} + 10\hat{k}, -\hat{i} - 3\hat{j} + 7\hat{k}, 5\hat{i} - \hat{j} + h\hat{k}$ and $7\hat{i} - 4\hat{j} + 7\hat{k}$ is $11$ cubic units then the value of $h$ is ______ ($h > 1$)

Let $\vec{a} = \hat{i} - \hat{j}, \vec{b} = \hat{i} + 2\hat{j} + 2\hat{k}, \vec{c} = -\hat{i} - \hat{j} + \hat{k}$ and $\vec{d} = 2\hat{i} - \hat{j} + \hat{k}$, then the shortest distance between the lines $\vec{r} = \vec{a} + t\vec{b}$ and $\vec{r} = \vec{c} + p\vec{d}$ is $k$, then the value of $\frac{1}{k^2}$ is ______.

In the above problem find the largest possible value of $|\vec{PQ}|$.

Let $\vec{a} = \vec{i} + 2\vec{j} + 3\vec{k}, \vec{b} = 2\vec{i} + 3\vec{j} + \vec{k}$ and $\vec{l} = \vec{i} + \vec{m}d$. If $(\vec{a} \times \vec{b}) = (\vec{a} \times \vec{c}) \times \vec{b}$. If $\vec{a} = \vec{0}$, then $|\vec{l}|$ is equal to

If $\begin{vmatrix} \vec{a} - 2\vec{b} + \vec{c} & \vec{b} + 2\vec{c} + 3\vec{a} \\ \vec{a} & \vec{b} & \vec{c} \\ \vec{b} & \vec{b} & \vec{c} \\ \vec{c} & \vec{c} & \vec{c} \end{vmatrix} = 54$, where $\vec{a}$, $\vec{b}$, $\vec{c}$ are 3 non-coplanar vectors, then the value of $[\vec{a} \, \vec{b} \, \vec{c}]$ is equal to

69. Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three unit vectors, out of which vectors $\vec{b}$ and $\vec{c}$ are non-parallel. If $\alpha$ and $\beta$ are the angles which vector $\vec{a}$ makes with vectors $\vec{b}$ and $\vec{c}$ respectively and $\vec{a} \times (\vec{b} \times \vec{c}) = \dfrac{1}{2}\vec{b}$, then $|\alpha - \beta|$ is equal to ________ °.

Given $\vec{a} \times (\vec{b} \times \vec{c}) = \dfrac{1}{2}\vec{b}$, where $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors. If $\alpha$ is the angle between $\vec{a}$ and $\vec{c}$ and $\beta$ is the angle between $\vec{a}$ and $\vec{b}$, find $|\alpha - \beta|$.

If $\vec{i} \times [(\vec{a} - \vec{j}) \times \vec{i}] + \vec{j} \times [(\vec{a} - \vec{k}) \times \vec{j}] + \vec{k} \times [(\vec{a} - \vec{i}) \times \vec{k}] = \vec{0}$ and $\vec{a} = x\vec{i} + y\vec{j} + z\vec{k}$, then the value of $8(x^3 - 2y + 2z)$ is equal to

Let $\vec{a}=-\hat{i}+\hat{j}+2\hat{k}$, $\vec{b}=\hat{i}-\hat{j}-3\hat{k}$, $\vec{c}=\vec{a}\times\vec{b}$ and $\vec{d}=\vec{c}\times\vec{a}$. Then $(\vec{a}-\vec{b})\cdot\vec{d}$ is equal to:

Let $\vec{a}=2\hat{i}-5\hat{j}+5\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+3\hat{k}$. If $\vec{c}$ is a vector such that $2(\vec{a}\times\vec{c})+3(\vec{b}\times\vec{c})=\vec{0}$ and $(\vec{a}-\vec{b})\cdot\vec{c}=-97$, then $|\vec{c}\times\hat{k}|^2$ is equal to:

Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$, $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\vec{a}\times\vec{b}$. Let $\vec{d}$ be a vector such that $|\vec{d}-\vec{a}|=\sqrt{11}$, $|\vec{c}\times\vec{d}|=3$ and the angle between $\vec{c}$ and $\vec{d}$ is $\dfrac{\pi}{4}$. Then $\vec{a}\cdot\vec{d}$ is equal to:

Let $\vec{r} = (\vec{a} \times \vec{b}) \sin x + (\vec{b} \times \vec{c}) \cos y + (\vec{c} \times \vec{a})$, where $\vec{a}$, $\vec{b}$ and $\vec{c}$ are non-zero non-coplanar vectors. If $\vec{r}$ is orthogonal to $3\vec{a} + 5\vec{b} + 2\vec{c}$, then the value of $\sec^2 y + \cosec^2 x + \sec y \cosec x$ is

63. Let $\vec{a} = 3\hat{i} + 2\hat{j} + x\hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$, for some real $x$. Then $|\vec{a} \times \vec{b}| = r$ is possible if:

Let the given points be A(−1, −1, 2), B(2, m, 5) and C(3, 11, 6). Find the value of m such that A, B and C are collinear.

Let $\vec{a}=\hat{i}+\hat{j}+2\hat{k}$, $\vec{b}=2\hat{i}-\hat{j}+\hat{k}$, $\vec{c}=3\hat{i}-\hat{k}$. A vector $\vec{v}$ is coplanar with $\vec{a}$ and $\vec{b}$, perpendicular to $\vec{c}$, and satisfies $\vec{v}\cdot(\hat{i}+2\hat{j}+\hat{k})=8$. Find $|\vec{v}|^2$.

If position vector of a point A is $\vec{a} + 2\vec{b}$ and any point P($\vec{a}$) divides AB in the ratio of 2:3, then position vector of B is

The vector a = αi + 2j + βk lies in the plane of the vectors b = i + j and c = j + k and bisects the angle between b and c. Then, which one of the following gives possible values of α and β?

If x̂, ŷ and ẑ are three unit vectors in three-dimensional space, then the minimum value of $|\hat{x}+\hat{y}|^2+|\hat{y}+\hat{z}|^2+|\hat{z}+\hat{x}|^2$ is

The co-ordinates of the point P on the line \[\mathbf{r} = (\mathbf{i} + \mathbf{j} + \mathbf{k}) + \lambda(-\mathbf{i} + \mathbf{j} - \mathbf{k})\] which is nearest to the origin is

[JEE Main 2021] If $|\vec{a}|=|\vec{b}|=|\vec{a}-\vec{b}|=1$, then $|\vec{a}+\vec{b}|$ is

Given the direction vectors of two skew lines $L_1$ and $L_2$ as $\hat{i}+\hat{j}$ and $\hat{j}+\hat{k}$ respectively, the angle between them satisfies $\cos\theta=$

If u and v are unit vectors and θ is the acute angle between them, then 2u × 3v is a unit vector for

Algebra Questions (626)