Vector Algebra Questions - JEE

If the unit vectors \mathbf{e}_1 and \mathbf{e}_2 are inclined at an angle 2\theta and |\mathbf{e}_1 - \mathbf{e}_2| , then for \theta \in [0, \pi], \theta may lie in the interval

If the vectors a = i − j + 2k, b = 2i + 4j + k and c = λi + j + μk are mutually orthogonal, then (λ, μ) is equal to

Let $\vec{m}$ and $\vec{n}$ are unit vectors and $\vec{w}$ is vector such that $\vec{m} \times \vec{n} + \vec{m} = \vec{w}$ and $\vec{w} \times \vec{m} = \vec{n}$. Then find the value of $[\vec{m}, \vec{n}, \vec{w}]$.

[JEE Main 2020] Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $\vec{a}\neq\vec{0},\;|\vec{b}|=4,\;|\vec{c}|=2$. Given $\vec{a}=\vec{b}\times(2\vec{a}+\lambda\vec{c})$, $\lambda>0$. If angle between $\vec{b}$ and $\vec{c}$ is $\pi/3$ and $(\vec{a}\times\vec{b})\cdot\vec{c}=|\vec{a}|$, then $\lambda$ equals

Let a vector $\hat{i}+\sqrt{2}\,\hat{j}+\sqrt{2}\,\hat{k}$ be obtained by rotating the vector $\sqrt{3}\,\hat{j}$ by an angle $45°$ about the origin in the clockwise direction to the first quadrant. Then the area of the triangle formed by the vector $(\hat{i}+\sqrt{2}\,\hat{j}+\sqrt{2}\,\hat{k})$ with the coordinate axes is equal to:

[JEE Main 2019] Let $\vec{a}=\hat{i}+2\hat{j}-\sqrt2\hat{k}$ and $\vec{b}=\sqrt2\hat{i}-\hat{j}+\sqrt2\hat{k}$. If $\vec{c}=\vec{a}\times(\vec{a}\times\vec{b})$, then $|\vec{c}|$ equals

Let $\hat{a},\hat{b},\hat{c}$ be three mutually perpendicular unit vectors and $\vec{d}=\lambda(\hat{a}+\hat{b}+\hat{c})$. If $|\vec{d}-\hat{a}|^2+|\vec{d}-\hat{b}|^2+|\vec{d}-\hat{c}|^2=8$, find $\lambda$.

[JEE Main 2020] Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $\vec{a}\neq\vec{0},\;|\vec{b}|=4,\;|\vec{c}|=2$. Given $\vec{a}=\vec{b}\times(2\vec{a}+\lambda\vec{c})$, $\lambda>0$. If angle between $\vec{b}$ and $\vec{c}$ is $\pi/3$ and $(\vec{a}\times\vec{b})\cdot\vec{c}=|\vec{a}|$, then $\lambda$ equals

Four points $A(1,-1,1),\;B(3,1,1),\;C(6,3,1)$ and $D(6,-1,-1)$ taken in order are the vertices of

Given that $\vec{u} = \hat{i} + \hat{j}$, $\vec{v} = \hat{i} - \hat{j}$, $\vec{\omega} = \hat{i} + 2\hat{j} + 3\hat{k}$. Let $\vec{n} = a\hat{i} + b\hat{j} + c\hat{k}$ be a unit vector such that $\vec{u} \cdot \vec{n} = 0$ and $\vec{v} \cdot \vec{n} = 0$. Find $|\vec{\omega} \cdot \vec{n}|$.

70. If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors, then $|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2$ does not exceed ________.

Given $\vec{a} = \alpha\hat{i} + \hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} + \hat{j} - \alpha\hat{k}$ and $\vec{c} = \alpha\hat{i} - 2\hat{j} + 3\hat{k}$ are coplanar. If $S$ is the set of all values of $\alpha$, then find $S$.

If $t_1$ and $t_2$ are the times of flight of two particles having the same initial velocity $u$ and range $R$ on the horizontal, then $t_1^2 + t_2^2$ is equal to

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$. A vector $\vec{b}$ satisfies $\vec{a}\cdot\vec{b}=|\vec{b}|^2$ and $|\vec{a}-\vec{b}|^2=7$. Find $|\vec{b}\times\vec{a}|^2$.

If two points $P$ and $Q$ are on the curve $y = 2^{x+1}$, such that $\overrightarrow{OP} \cdot \vec{i} = -1$ and $\overrightarrow{OQ} \cdot \vec{i} = 2$, where $\vec{i}$ is a unit vector along the $x$-axis, then $|\overrightarrow{OQ} - \overrightarrow{OP}|$ is equal to

Here, m = cos(π/4) = 1/√2 and n = cos(π/2) = 0. If l² + m² + n² = 1, find the value of l.

Let the height of a triangle be l, where a triangle has a base of 5 units. Points are given as A(1, -1, 2), B(-2, 1, 0) (direction cosine of the line), and C(3, 0, 4). Find the area of the triangle (in square units, rounded to 3 decimal places).

Let $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. A vector $\vec{c}$ satisfies $\vec{a}\times\vec{c}=\vec{b}$ and $\vec{a}\cdot\vec{c}=3$. Find $|\vec{c}|^2$.

With two forces acting at a point, the maximum effect is obtained when their resultant is 4 N. If they act at right angles, their resultant is 3 N. Then the factors are

The projections of a vector on the three coordinate axes are 6, $-3$, 2, respectively. The direction cosines of the vector are

Vector coplanar with $\vec{a} = \hat{i} - \hat{j}$ and $\vec{b} = \hat{i} + 2\hat{j}$ is given by

If the vectors $\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}$ and such that $\vec{a}$, $\vec{c}$ and $\vec{b}$ form a right handed system, then $\vec{c}$ is

Given that $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors. We know that $|\vec{a}+\vec{b}+\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}) \geq 0$. Find the minimum value of $|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2$.

If a unit vector $\vec{a}$ makes angles $\pi/3$ with $\hat{i}$, $\pi/4$ with $\hat{j}$ and $\theta \in (0, \pi)$ with $\hat{k}$, then a value of $\theta$ is ______ (in degrees).

The vectors $\overrightarrow{AB} = 3\hat{i} + 4\hat{k}$ and $\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of a triangle $ABC$. The length of the median through $A$ is

In a right angle $\triangle ABC$, $\angle A = 90°$ and sides $a, b, c$ are, respectively, 5 cm, 4 cm and 3 cm. If a force $\vec{F}$ has moments 0, 9 and 16 in N cm units, respectively, about vertices $A$, $B$ and $C$, then magnitude of $\vec{F}$ is

Let $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. If $\vec{c}=\alpha\vec{a}+\beta\vec{b}$ satisfies $\vec{c}\times\vec{a}=\vec{b}$, find $\alpha+\beta$.

Let 61. Let $\vec{a} = 2\hat{i} + \lambda_1\hat{j} + 3\hat{k}$, $\vec{b} = 4\hat{i} + (3-\lambda_2)\hat{j} + 6\hat{k}$ and $\vec{c} = 3\hat{i} + 6\hat{j} + (\lambda_3 - 1)\hat{k}$ be three vectors such that $\vec{b} = 2\vec{a}$ and $\vec{a}$ is perpendicular to $\vec{c}$. Then a possible value of $(\lambda_1, \lambda_2, \lambda_3)$ is:

A tangent is drawn to the curve $y=x^2$ at a point $A(x_1,y_1)$. The scalar product $\overrightarrow{AR}\cdot\overrightarrow{AP}$ at point $P(a,a^2)$ equals (where $A$ is any point on the curve)

Let $\vec{a}=q_1\hat{i}+q_2\hat{j}+q_3\hat{k}$ make equal angles with OX, OY, OZ and $|\vec{a}|=\sqrt{3}$. If the projection of $\vec{a}$ on $\hat{i}+\hat{j}-\hat{k}$ is 1, find $q_1+q_2+q_3$.

The distance of the point having position vector 2i + 6j + 3k from the straight line passing through the point (2, 3, 4) and parallel to the vector i + 4j - 6k is

The projection of the vector $\vec{i} - \vec{j}$ on the vector $\vec{i} + \vec{j}$ is

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be non-zero vectors such that $(\vec{a}\times\vec{b})\times\vec{c} = \dfrac{1}{3}|\vec{b}||\vec{c}|\vec{a}$. If $\theta$ is the acute angle between the vectors $\vec{b}$ and $\vec{c}$, then $\sin\theta$ equals

For a triangle ABC, let $\vec{p}=\overrightarrow{BC}$, $\vec{q}=\overrightarrow{CA}$ and $\vec{r}=\overrightarrow{BA}$. If $|\vec{p}|=2\sqrt{3}$, $|\vec{q}|=2$ and $\cos\theta=\dfrac{1}{\sqrt{3}}$, where $\theta$ is the angle between $\vec{p}$ and $\vec{q}$, then $|\vec{p}\times(\vec{q}-3\vec{r})|^2+3|\vec{r}|^2$ is equal to:

Let $PQR$ be a triangle such that $\overrightarrow{PQ}=-2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{PR}=a\hat{i}+b\hat{j}-4\hat{k}$, $a,b\in\mathbb{Z}$. Let $S$ be the point on QR which is equidistant from the lines PQ and PR. If $|\overrightarrow{PR}|=9$ and $\overrightarrow{PS}=\hat{i}-7\hat{j}+2\hat{k}$, then the value of $3a-4b$ is ___.

Direction vectors of two lines $L_1$ and $L_2$ are $\hat{i}-\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}-\hat{k}$ respectively. The angle between the lines is

A, B, C, D are four points in space and satisfy $|\vec{AB}| = 3, |\vec{BC}| = 7, |\vec{CD}| = 11$ and $|\vec{DA}| = 9$. Then find the value of $\vec{AC} \times \vec{BD}$.

The scalar product of the vector $\hat{i} + \hat{j} + \hat{k}$ with a unit vector along the sum of vectors $2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\lambda\hat{i} + 2\hat{j} + 3\hat{k}$ is equal to one. The value of $\lambda$ is

Find $\frac{\Delta_2}{\Delta}$

Let $\vec{a}=-5\hat{i}+\hat{j}-3\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-4\hat{k}$. Let $\vec{c}=\left(\left(\left(\vec{a}\times\vec{b}\right)\times\hat{i}\right)\times\hat{i}\right)\times\hat{i}$. Then $\vec{c}\cdot(-\hat{i}+\hat{j}+\hat{k})$ is equal to:

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If $\vec{a}+5\vec{b}$ is collinear with $\vec{c}$, $\vec{b}+6\vec{c}$ is collinear with $\vec{a}$, and $\vec{a}+\alpha\vec{b}+\beta\vec{c}=\vec{0}$, then $\alpha+\beta$ is equal to:

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

Vector Algebra Questions (573)