Algebra Questions - JEE

Let PQRS be a parallelogram whose diagonals $\overrightarrow{PR}=3\hat{i}+\hat{j}$ and $\overrightarrow{QS}=\hat{i}-\hat{j}+\hat{k}$. Find the area of PQRS.

[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

259. Given 2019 vectors on a plane. Sum of every 2018 vectors is a scalar multiple of other vector. Not all vectors are scalar multiple of each other. The magnitude of sum of all these vectors is:

Example 33. The vector $\mathbf{c}$, directed along the internal bisector of the angle between the vectors $\mathbf{a} = 7\mathbf{i} - 4\mathbf{j} - 4\mathbf{k}$ and $\mathbf{b} = -2\mathbf{i} - \mathbf{j} + 2\mathbf{k}$ with $|\mathbf{c}| = 5\sqrt{6}$, is

If the vector $6\hat{i}-3\hat{j}-6\hat{k}$ is decomposed into vectors parallel and perpendicular to the vector $\hat{i}+\hat{j}+\hat{k}$, then the two vectors are

If $\vec{u}$, $\vec{v}$ and $\vec{w}$ are three non-coplanar vectors, then $(\vec{u}+\vec{v}-\vec{w})\cdot(\vec{u}-\vec{v})\times(\vec{v}-\vec{w})$ equals

Given $ \overrightarrow{OC} = m\overrightarrow{OA} + n\overrightarrow{OB} $, i.e., $ \vec{c} = m\vec{a} + n\vec{b} $, where $ |\vec{a}| = 1,\ |\vec{b}| = 1,\ |\vec{c}| = \sqrt{2},\ \tan\alpha = 7 $. Find the value of $ m + n $ (or the relevant expression as given in the problem).

If $\hat{u}$ and $\hat{v}$ are unit vectors and $\theta$ is the acute angle between them, then $2\hat{u}\times 3\hat{v}$ is a unit vector for

If $|\vec{a}| = 2$, $|\vec{b}| = 3$ and $|2\vec{a} - \vec{b}| = 5$, then $|2\vec{a} + \vec{b}|$ equals

Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ is the angle between them. Then $\vec{a} + \vec{b}$ is a unit vector, if

Let OA = a, OB = 10a + 2b and OC = b, where O, A and C are non-collinear points. Let p denote the area of quadrilateral OACB, and let q denote the area of parallelogram with OA and OC as adjacent sides. If p = kq, then k is equal to

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 $\alpha+\beta$.

Three forces $\vec{P}$, $\vec{Q}$ and $\vec{R}$ acting along $IA$, $IB$ and $IC$, where $I$ is the incentre of a $\triangle ABC$, are in equilibrium. Then $\vec{P} : \vec{Q} : \vec{R}$ is

The resultant of two forces P N and 3 N is a force of 7 N. If the direction of 3 N force were reversed, the resultant would be $\sqrt{19}$ N. The value of P is

The vectors $\vec{a} = 2\lambda \vec{i} + 4\lambda \vec{j} + \vec{k}$ and $\vec{b} = 7\vec{i} - 2\vec{j} + \lambda \vec{k}$ make an obtuse angle whereas the angle between $\vec{b}$ and $\vec{k}$ is acute and less than $\pi/6$. Find the range of $\lambda$.

In a triangle ABC, right-angled at the vertex A, if the position vectors of A, B and C are, respectively, $3\hat{i}+\hat{j}-\hat{k}$, $-\hat{i}+3\hat{j}+p\hat{k}$ and $5\hat{i}+q\hat{j}-4\hat{k}$, then the point $(p, q)$ lies on a line

Let $\vec{a}=4\hat{i}-\hat{j}+\hat{k}$, $\vec{b}=11\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b})\times\vec{c}=\vec{c}\times(-2\vec{a}+3\vec{b})$. If $(2\vec{a}+3\vec{b})\cdot\vec{c}=1670$, then $|\vec{c}|^2$ is equal to:

Let a = î + ĵ + k̂, b = î − ĵ + 2k̂ and c = xî + (x−2)ĵ − k̂. If the vector c lies in the plane of a and b, then x equals

[JEE Main 2021] Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. Let $\vec{c}$ be a vector such that $\vec{a}\cdot\vec{c}=|\vec{c}|$, $|\vec{c}-\vec{a}|=2\sqrt2$ and the angle between $\vec{a}\times\vec{b}$ and $\vec{c}$ is $\dfrac\pi6$. Then $|(\vec{a}\times\vec{b})\times\vec{c}|$ equals

Given $\vec{a} = \hat{i} - \hat{j}$, $\vec{b} = \hat{i} + \hat{j} + \hat{k}$, $\vec{a} \times \vec{c} + \vec{b} = \vec{0}$ and $\vec{a} \cdot \vec{c} = 4$. Find $|\vec{c}|^2$.

For $p>0$, the vector $\vec{v}_2=(2,\,-(\sqrt{3}\,p+1))$ is obtained by rotating $\vec{v}_1=\sqrt{3}(-1,\,-(p^2+\sqrt{3}))$ about the origin counter-clockwise. If the angle of rotation is $\theta$, find $\tan\theta$.

The resultant of forces $\vec{P}$ and $\vec{Q}$ is $\vec{R}$. If $\vec{Q}$ is doubled then $\vec{R}$ is doubled. If the direction of $\vec{Q}$ is reversed, then $\vec{R}$ is again doubled. Then $P^2 : Q^2 : R^2$ is

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\vec{c} = \hat{a} + 2\hat{b}$ and $\vec{d} = 5\hat{a} - 4\hat{b}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is

71. If $\vec{a}$, $\vec{b}$ and $\vec{c}$ 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 $\pi/3$, then the value of $|\vec{a} \times \vec{b} - \vec{a} \times \vec{c}|$ is ________.

Example 30. Let A, B and C be unit vectors. Suppose A · B = A · C = 0 and the angle between B and C is π/4. Then:

We have $\vec{b} \times \vec{c} = \vec{b} \times \vec{d}$. Then $\vec{d}$ equals:

72. Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 4\hat{i} + 3\hat{j} + 4\hat{k}$ and $\vec{c} = \hat{i} + \alpha\hat{j} + \beta\hat{k}$ are linearly dependent and $|\vec{c}| = \sqrt{3}$, then $|\alpha| + \beta =$ ________.

The area (in sq. units) of the parallelogram, whose diagonals are along the vectors $8\hat{i}-6\hat{j}$ and $3\hat{i}+4\hat{j}-12\hat{k}$, is

[JEE Main 2021] Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. Let $\vec{c}$ be a vector such that $\vec{a}\cdot\vec{c}=|\vec{c}|$, $|\vec{c}-\vec{a}|=2\sqrt2$ and the angle between $\vec{a}\times\vec{b}$ and $\vec{c}$ is $\dfrac\pi6$. Then $|(\vec{a}\times\vec{b})\times\vec{c}|$ equals

Let three vectors $\vec{a},\vec{b},\vec{c}$ satisfy $\vec{a}\times\vec{b}=\vec{c}$ and $\vec{b}\times\vec{c}=\vec{a}$, with $|\vec{a}|=1$. If the angle between $\vec{b}$ and $\vec{c}$ is $\dfrac{\pi}{6}$, find $|\vec{b}|$.

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}+\alpha\hat{j}+\hat{k}$, $\vec{c}=\hat{i}-4\hat{j}+5\hat{k}$. If the volume of the parallelepiped with adjacent sides $\vec{a},\vec{b},\vec{c}$ is 2, find the value of $6\alpha$.

Let PQRS be a parallelogram whose diagonals $\overrightarrow{PR}=3\hat{i}+\hat{j}$ and $\overrightarrow{QS}=\hat{i}-\hat{j}+\hat{k}$. Find the area of PQRS.

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 $|\vec{c}|^2$.

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. If $\hat{c}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$, find $|\hat{c}\cdot(3\hat{i}-\hat{j}+\hat{k})|$.

Let $\vec{a}=2\hat{i}+3\hat{j}-\hat{k}$, $\vec{b}=\hat{i}-\hat{j}+2\hat{k}$, $\vec{c}=2\hat{i}+\hat{j}-\hat{k}$. If $[\vec{a}+\vec{b},\,\vec{b}+\vec{c},\,\vec{c}+\vec{a}]=\lambda[\vec{a},\vec{b},\vec{c}]$, find $\lambda$.

Let $OAB$ be a regular triangle (equilateral) with $O$ at the origin. If $\overrightarrow{OA}=\vec{a}$ and $\overrightarrow{OB}=\vec{b}$, then which of the following hold?

Given three vectors $\vec{U}=\hat{i}+\hat{j}+\hat{k}$, $\vec{V}=\hat{i}+\hat{j}-\hat{k}$, $\vec{W}=\hat{i}-\hat{j}+\hat{k}$. Which of the following hold?

Consider the regular hexagon ABCDEF with centre at O (origin). Find $\vec{AD} + \vec{EB} + \vec{FC}$

From the parallelogram law of forces, if $R^2 = P^2 + Q^2 + 2PQ\cos\alpha$ and equations are set up for doubled forces, the ratio $P^2 : Q^2 : R^2$ equals:

If ABCD is a parallelogram and the position vectors of A, B and C are $\vec{i} + 3\vec{j} + 5\vec{k}$, $\vec{i} + \vec{j} + \vec{k}$ and $7\vec{i} + 7\vec{j} + 7\vec{k}$, then the position vector of D will be

The horizontal force and the force inclined at an angle 60° with the vertical, whose resultant is in vertical direction of P kg, are

If the vectors $6\mathbf{i}, 2\mathbf{j}$ and $3\mathbf{i} + 6\mathbf{j} - 2\mathbf{k}$ form a triangle, determine the type of triangle.

Let a = 3i − 6j + k and b = 2i − 4j + lk. Since a and b are parallel, find the value of l.

If position vector of point A is a + 2b and a divides AB in the ratio 2 : 3, then the position vector of B is

If a and b are the position vectors of A and B respectively, then the position vector of a point C on AB produced such that AC = 3 AB is

We have a, b, c are non-zero vectors and $\vec{a} \neq \lambda_1 \vec{b}$ and $\vec{b} \neq \mu_1 \vec{c}$, $\lambda_1, \mu_1 \neq 0$. Given:$\vec{a} + 2\vec{b} = m\vec{c}$ ...(1)and $\vec{b} + 3\vec{c} = n\vec{a}$ ...(2)where $m, n \neq 0$ are reals. Find $\vec{a} + 2\vec{b} + 6\vec{c}$ = ?

If $a$, $b$ and $c$ are $p$th, $q$th, $r$th terms of HP and $\vec{u} = (q - r)\vec{i} + (r - p)\vec{j} + (p - q)\vec{k}$, $\vec{v} = \frac{1}{a}\vec{i} + \frac{1}{b}\vec{j} + \frac{1}{c}\vec{k}$, then

The angles of a triangle, two of whose sides are represented by vectors 3(a × b) and b - (a · b)a, where b is a non-zero vector and a is a unit vector in the direction of a, are

Direction of the ant's resultant displacement after first three steps is :

If the position vectors of the points $A$, $B$ and $C$ be $\vec{i} + \vec{j}$, $\vec{i} - \vec{j}$ and $a\vec{i} + b\vec{j} + c\vec{k}$ respectively, then the points $A$, $B$ and $C$ are collinear, if

Algebra Questions (626)