Vector Algebra Questions - JEE

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

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?

Vector Algebra Questions (573)