Algebra Questions (626)

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

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 \(\cos\theta=\)

Let \(\vec{a}=2\hat{i}-\hat{j}+4\hat{k}\) and \(\vec{b}=\hat{i}+\alpha\hat{j}+\beta\hat{k}\). If \(\vec{b}\) is perpendicular to \(3\hat{i}-4\hat{j}+\hat{k}\) and the projection of \(\vec{b}\) on \(\vec{a}\) is \(\dfrac{17}{\sqrt{21}}\), find \(|\vec{b}|\).

Let $\vec{U} = \vec{i} + \vec{j}$, $\vec{V} = \vec{i} - \vec{j}$ and $\vec{W} = 3\vec{i} + 5\vec{j} + 3\vec{k}$. If $\vec{n}$ is a unit vector such that $\vec{U} \cdot \vec{n} = 0$ and $\vec{V} \cdot \vec{n} = 0$, then $|\vec{W} \cdot \vec{n}|$ is equal to

Let the volume of a parallelepiped whose coterminous edges are given by $\vec{u} = \vec{i} + \vec{j} + \vec{k}$, $\vec{v} = \vec{i} + \vec{j} + 3\vec{k}$ and $\vec{w} = 2\vec{i} + \vec{j} + \vec{k}$ be 1 cu. unit. If $\theta$ is the angle between the edges $\vec{u}$ and $\vec{w}$, then the value of $\cos \theta$ can be

67. Let \(\vec{\alpha} = (\lambda - 2)\vec{a} + \vec{b}\) and \(\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}\) be two given vectors where vectors \(\vec{a}\) and \(\vec{b}\) are non-collinear. The value of \(|\lambda|\) for which vectors \(\vec{\alpha}\) and \(\vec{\beta}\) are collinear, is ________.

$|\vec{a}| = |4\vec{b} - \lambda^2\vec{c}|$, $|\vec{a}|^2 = 10|\vec{b}|^2 + |\vec{c}|^2 - 8\vec{b}.\vec{c}$, $9 = 16 + \lambda^2 - 8\vec{b}.\vec{c}$, and $\lambda^2 - 8\lambda + 7 = 0$ (as $\vec{b}.\vec{c} = 1$). Find the sum of all values of $\lambda = 8$.

Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be ^i + 2^j + k, ^ ^ i + 3 j - 2k and ^ ^ ^ ^ ^ 2i + j - k respectively. The altitude from the vertex D to the opposite face ABC meets the median line \sqrt110 segment through A of the triangle ABC at the point E. If the length of AD is 3 and the volume of the \sqrt805 tetrahedron is , then the position vector of E is 6\sqrt2

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?

As \(|\vec{a}| = 1\), \(|\vec{b}| = 1\), \(|\vec{a}+\vec{b}| = \sqrt{3}\), \(\vec{c} = \vec{a} + 2\vec{b} + 3(\vec{a} \times \vec{b})\). Find \(2|\vec{c}|\).

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 \(|\vec{c} - \vec{a}| = 3\), \(|(\vec{a} \times \vec{b}) \times \vec{c}| = 3\) and the angle between \(\vec{c}\) and \(\vec{a} \times \vec{b}\) be \(30°\). Then \(\vec{a} \cdot \vec{c}\) is equal to

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).