Vector Algebra Questions - JEE

If the vertices A, B, C of a △ABC have position vectors $(1, 2, 3)$, $(-1, 0, 0)$, $(0, 1, 2)$ respectively, then $\angle ABC$ (the angle between the vectors $\vec{BA}$ and $\vec{BC}$) is equal to

Given $\sqrt{3}\hat{i}+\hat{j}$, $\hat{i}+\sqrt{3}\hat{j}$ and $\beta\hat{i}+(1-\beta)\hat{j}$ respectively be the position vectors of the points A, B and C with respect to the origin O. If the angle bisector of $\angle AOB$ passes through C, find the sum of all possible values of $\beta$.

Let $\overrightarrow{OA}=2\vec{a}$, $\overrightarrow{OB}=6\vec{a}+5\vec{b}$ and $\overrightarrow{OC}=3\vec{b}$, where $O$ is the origin. If the area of the parallelogram with adjacent sides $\overrightarrow{OA}$ and $\overrightarrow{OC}$ is 15 sq. units, then the area (in sq. units) of the quadrilateral $OABC$ is equal to:

An arc PQ of a circle subtends a right angle at its centre O. The midpoint of the arc PQ is R. If $\overrightarrow{OP}=\vec{a}$ and $\overrightarrow{OQ}=\vec{b}$, find $\overrightarrow{OR}$.

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:

Given $\vec{a} = 2\hat{i} + \lambda_1 \hat{j} + 3\hat{k}$, $\vec{b} = 4\hat{i} + (3 - \lambda_2)\hat{j} + 6\hat{k}$ and $\vec{c} = 3\hat{i} + 6\hat{j} + (\lambda_3 - 1)\hat{k}$, and $\vec{b} = 2\vec{a}$, $\vec{a}$ is perpendicular to $\vec{c}$. Find the values of $(\lambda_1, \lambda_2, \lambda_3)$.

If the position vectors of the vertices of a triangle be $2\hat{i} + 4\hat{j} - \hat{k}$, $4\hat{i} + 5\hat{j} + \hat{k}$ and $3\hat{i} + 6\hat{j} - 3\hat{k}$, then the triangle is

If a, b and c are non-coplanar vectors and $\lambda$ is a real number, then the vectors $\mathbf{a} + 2\mathbf{b} + 3\mathbf{c}$, $\lambda\mathbf{b} + 4\mathbf{c}$ and $(2\lambda - 1)\mathbf{c}$ are non-coplanar for

Let $\vec{u} = \hat{i}+\hat{j}$, $\vec{v} = \hat{i}-\hat{j}$ and $\vec{w} = \hat{i}+2\hat{j}+3\hat{k}$. If $\hat{n}$ is unit vector such that $\vec{u}\cdot\hat{n} = 0$ and $\vec{v}\cdot\hat{n} = 0$, then $|\vec{w}\cdot\hat{n}|$ is equal to

For some non-zero vector V, if the sum of V and the vector obtained from V by rotating it by ∠2α equals to the vector obtained from V by rotating it by ∠α, then the value of α, is

Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity $\vec{u}$ and the other from rest with uniform acceleration $\vec{f}$. Let $\alpha$ be the angle between their directions of motion. The relative velocity of the second particle with respect to the first is least after a time

65. Let $\vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}$ be two vectors. If a vector perpendicular to both the vectors $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ has the magnitude 12, then one such vector is:

Two particles start from the same point. The 1st particle moves with uniform velocity $u$ and the 2nd particle starts from rest with uniform acceleration $f$. The angle between their directions of motion is $\alpha$. The relative velocity of the 2nd particle with respect to 1st is $R = \sqrt{f^2t^2 + 4^2 - 2ftu\cos\alpha}$. For the least value of $R$ (relative velocity), $\dfrac{dR}{dt} = 0$. The time at which the relative velocity is minimum is

[JEE Main 2021] Suppose $\vec{a},\vec{b},\vec{c}$ are unit vectors and $(\vec{a}+3\vec{b})\perp\vec{c}$ and $(\vec{a}+\vec{b})\perp(\vec{a}+3\vec{b})$. Then which of the following is true?

Given $\vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}$. A vector perpendicular to both $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ is

If the volume of parallelopiped formed by the vectors $\hat{i} + \lambda\hat{j} + \hat{k}$, $\hat{j} + \lambda\hat{k}$ and $\lambda\hat{i} + \hat{k}$ is minimum, then $\lambda$ is equal to ______ (up to three decimal places).

Given two vectors are $\hat{i} - \hat{j}$ and $\hat{i} + 2\hat{j}$ the unit vector coplanar with the two vectors and perpendicular to first is

A particle acted on by constant forces $4\hat{i}+\hat{j}-3\hat{k}$ and $3\hat{i}+\hat{j}-\hat{k}$ is displaced from the point $\hat{i}+2\hat{j}+3\hat{k}$ to the point $5\hat{i}+4\hat{j}+\hat{k}$. The total work done by the forces is

If $(\vec{a}\times\vec{b})\times\vec{c} = \vec{a}\times(\vec{b}\times\vec{c})$, where $\vec{a}$, $\vec{b}$ and $\vec{c}$ are any three vectors such that $\vec{a}\cdot\vec{b}\neq 0$, $\vec{b}\cdot\vec{c}\neq 0$, then $\vec{a}$ and $\vec{c}$ are

For a non-zero vector $\mathbf{a}$, the set of real numbers satisfying \(|(5-x)\mathbf{a}|

Let $\vec{a},\, \vec{b},\, \vec{c}$ be three vectors of magnitude 2, 3, 5 respectively, satisfying $|[\vec{a},\, \vec{b},\, \vec{c}]| = 30$. If $(2\vec{a} + \vec{b} + \vec{c}) \cdot ((\vec{a} \times \vec{c}) \times (\vec{a} - \vec{c}) + \vec{b}) = k$, then the value of $\left(\dfrac{k}{103}\right)$ is:

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number, then the vectors $\vec{a} + 2\vec{b} + 3\vec{c}$, $\lambda\vec{b} + 4\vec{c}$ and $(2\lambda - 1)\vec{c}$ are non-coplanar for

If the vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ from the sides $BC$, $CA$ and $AB$, respectively, of a triangle $ABC$, then

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $|\vec{a}+\vec{b}|=\sqrt{3}$. If $\vec{c}=\vec{a}+2\vec{b}+3(\vec{a}\times\vec{b})$, then $2|\vec{c}|$ is equal to

Ex. 62: Let ABC be a triangle whose centroid is G, orthocentre is H and circumcentre is the origin 'O'. If D is any point in the plane of the triangle such that no three of O, A, C and D are collinear satisfying the relation $\vec{AD} + \vec{BD} + \vec{CH} + 3\vec{HG} = \lambda \vec{HD}$, then what is the value of the scalar $\lambda$?

64. Let $\vec{\alpha} = 3\hat{i} + \hat{j}$ and $\vec{\beta} = 2\hat{i} - \hat{j} + 3\hat{k}$. If $\vec{\beta} = \vec{\beta}_1 - \vec{\beta}_2$, where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$, then $\vec{\beta}_1 \times \vec{\beta}_2$ is equal to:

If $|\vec{a}| = 1$, $|\vec{b}| = 3$ and $|\vec{c}| = 5$, then the value of $[\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{c} - \vec{a}]$ is

Both the lines pass through origin. Line L1 is parallel to the vector $\vec{V}_1 = (\cos\theta + \sqrt{3})\,\hat{i} + (\sqrt{2}\sin\theta)\,\hat{j} + (\cos\theta - \sqrt{3})\,\hat{k}$ and L2 is parallel to the vector $\vec{V}_2 = a\hat{i} + b\hat{j} + c\hat{k}$ If the angle $\alpha$ between the lines is independent of $\theta$, find $\alpha$.

Given 59. $\vec{a} = \hat{i} + \hat{j} + \hat{k},\ \vec{c} = \hat{j} - \hat{k},\ \vec{a} \cdot \vec{b} = 3$ and $\vec{a} \times \vec{b} = \vec{c}$, find $|\vec{b}|$.

Given $\vec{\alpha} = 3\hat{i} + \hat{j}$, $\vec{\beta} = 2\hat{i} - \hat{j} + 3\hat{k}$. If $\vec{\beta} = \vec{\beta}_1 - \vec{\beta}_2$ and $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$, then $\vec{\beta}_1 \times \vec{\beta}_2$ is equal to

If a = 3i - 2j + k, b = 2i - 4j - 3k and c = -i + 2j + 2k, then a + b + c is

Ex. 63: Let a, b and c be unit vectors such that $\vec{a} + \vec{b} - \vec{c} = \vec{0}$. If the area of triangle formed by vectors $\vec{a}$ and $\vec{b}$ is A, then what is the value of $16A^2$?

The moment about the point $\vec{i} + 2\vec{j} + 3\vec{k}$ of a force represented by $\vec{i} + \vec{j} + \vec{k}$ acting through the point $2\vec{i} + 3\vec{j} + \vec{k}$, is

Example 30. The median AD of the △ABC is bisected at E. BE meets AC in F. Then, AF : AC is equal to

Consider the set of eight vectors V = {ai + bj + ck : a, b, c ∈ {−1, 1}}. Three non-coplanar vectors can be chosen from V in $2^p$ ways. Then p is ________.

A couple is of moment $\vec{G}$ and the force forming the couple is $\vec{P}$. If $\vec{P}$ is turned through a right angle, the moment of the couple thus formed is $\vec{H}$. If instead, the forces $\vec{P}$ are turned through an angle $\alpha$, then the moment of couple becomes

If |a| = |b| = |c| = 2 and $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{c} = \mathbf{c} \cdot \mathbf{a} = 2$, then $[\mathbf{a} \mathbf{b} \mathbf{c}] \cos 45°$ is equal to:

If C is the midpoint of AB and P is any point outside AB, then

Given $\vec{a} = \hat{i} + \hat{j} + \sqrt{2}\hat{k}$, $\vec{b} = b_1\hat{i} + b_2\hat{j} + \sqrt{2}\hat{k}$ and $\vec{c} = 5\hat{i} + \hat{j} + \sqrt{2}\hat{k}$. Projection of $\vec{b}$ on $\vec{a}$ is $|\vec{a}|$ and $(\vec{a}+\vec{b})$ is perpendicular to $\vec{c}$. Find $|\vec{b}|$.

Let a, b, c be unit vectors such that the Gram determinant \[\begin{vmatrix} \vec{a}\cdot\vec{a} & \vec{a}\cdot\vec{b} & \vec{a}\cdot\vec{c} \\ \vec{a}\cdot\vec{b} & \vec{b}\cdot\vec{b} & \vec{b}\cdot\vec{c} \\ \vec{a}\cdot\vec{c} & \vec{c}\cdot\vec{b} & \vec{c}\cdot\vec{c} \end{vmatrix} = [\vec{a}\;\vec{b}\;\vec{c}]^2 = 4.\] Find $[\vec{a}\;\vec{b}\;\vec{c}]$.

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

Vector Algebra Questions (573)