Vector Algebra Questions (573)

$\vec{d}\perp\vec{a}=2\hat{i}+7\hat{j}-\hat{k}$ and $\vec{b}=3\hat{i}+5\hat{k}$, $\vec{c}\cdot\vec{d}=12$. Then $(-\hat{i}+\hat{j}-\hat{k})\cdot(\vec{c}\times\vec{d})$ is equal to

Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC , divides the arc AC - -\to - -\to - -\to length of arc AB such that length of arc BC = 1 5 , and OC = \alphaOA + \betaOB, then \alpha + \sqrt2(\sqrt3 - 1)\beta is equal to

The vectors x\mathbf{i} + (x+1)\mathbf{j} + (x+2)\mathbf{k}, (x+3)\mathbf{i} + (x+4)\mathbf{j} + (x+5)\mathbf{k} and (x+6)\mathbf{i} + (x+7)\mathbf{j} + (x+8)\mathbf{k} are coplanar if x is equal to

Let a \pi 3 \to + 2\tob and 3a . If \lambdaa \to - \lambda\tob are perpendicular to each other, then the number of values of \lambda in [-1, 3] is :

If A, B, C, D and E are five coplanar points, then \(\vec{DA} + \vec{DB} + \vec{DC} + \vec{AE} + \vec{BE} + \vec{CE}\) is equal to

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a} = \alpha\vec{b} - \hat{n}$, $(\alpha \neq 0)$ and $\vec{b}\cdot\vec{c} = 12$, then $|\vec{c}\times(\vec{a}\times\vec{b})|$ is equal to:

Let \(\vec{a} = 2\hat{i}+\hat{j}-2\hat{k}\) and \(\vec{b} = \hat{i}+\hat{j}\). If \(\vec{c}\) is a vector such that \(\vec{a}\cdot\vec{c} = |\vec{c}|\), \(|\vec{c}-\vec{a}| = 2\sqrt{2}\) and the angle between \((\vec{a}\times\vec{b})\) and \(\vec{c}\) is 30°, then \(|(\vec{a}\times\vec{b})\times\vec{c}|\) is equal to ________.

Let $A(2\hat{i}+3\hat{j}+5\hat{k}), B(-\hat{i}+3\hat{j}+2\hat{k})$ and $C(\hat{i}+5\hat{j}+\mu\hat{k})$ are vertices of a triangle and its median through $A$ is equally inclined to the positive directions of the axes. Find the value of $2\lambda - \mu$

[JEE Main 2021] If \(|\vec{a}|=|\vec{b}|=|\vec{a}-\vec{b}|=1\), then \(|\vec{a}+\vec{b}|\) is

$\vec{a}$ and $\vec{b}$ are two non-collinear vectors then the points with position vectors $l_1\vec{a} + m_1\vec{b}, l_2\vec{a} + m_2\vec{b}, l_3\vec{a} + m_3\vec{b}$ are collinear then find the value of $\begin{vmatrix} 1 & 1 & 1 \\ l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{vmatrix}$.

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|2\vec{a}+3\vec{b}|=|3\vec{a}+\vec{b}|\). Find the angle between \(\vec{a}\) and \(\vec{b}\).

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 \(\overrightarrow{OR}\).

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\dfrac{\pi}{3}$. If $\lambda\hat{a}+2\hat{b}$ and $3\hat{a}-\lambda\hat{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1,3]$ is:

If \(\vec{x} = 3\hat{i} - 6\hat{j} - \hat{k}\), \(\vec{y} = \hat{i} + 4\hat{j} - 3\hat{k}\) and \(\vec{z} = 3\hat{i} - 4\hat{j} - 12\hat{k}\), then the magnitude of the projection of \(\vec{x} \times \vec{y}\) on \(\vec{z}\) is

The vector \(\hat{i} + x\hat{j} + 3\hat{k}\) is rotated through an angle \(\theta\) and doubled in magnitude, then it becomes \(4\hat{i} + (4x-2)\hat{j} + 2\hat{k}\). The values of \(x\) are

Let a ^ be a unit vector perpendicular to the vectors b = ^ \to i - 2 j + 3k and c = 2 i + 3 j - k, and makes an angle ^ ^ ^ ^ ^ of cos -1 (- 1 3 ) with the vector ^i + ^j + k ^ . If ^ a makes an angle of \pi 3 with the vector ^i + \alpha^j + k ^ , then the value of \alpha is :

If the position vectors of the vertices A, B and C of a \(\triangle ABC\) are, respectively, \(4\hat{i}+7\hat{j}+8\hat{k}\), \(2\hat{i}+3\hat{j}+4\hat{k}\) and \(2\hat{i}+5\hat{j}+7\hat{k}\), then the position vector of the point, where the bisector of \(\angle A\) meets BC is

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 \(\left\lfloor \dfrac{k}{103} \right\rfloor\) is:

Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda\hat{i}+4\hat{k}$, $\lambda>0$, on the vector $\vec{a}=\hat{i}+2\hat{j}+2\hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.

Let the position vectors of three vertices of a triangle be $4\vec{p}+\vec{q}-3\vec{r}$, $-5\vec{p}+\vec{q}+2\vec{r}$ and $2\vec{p}-\vec{q}+2\vec{r}$. If the position vectors of the orthocentre and the circumcentre of the triangle are $\dfrac{\vec{p}+\vec{q}+\vec{r}}{4}$ and $\alpha\vec{p}+\beta\vec{q}+\gamma\vec{r}$ respectively, then $\alpha+2\beta+5\gamma$ is equal to:

If the plane faces of a tetrahedon are represented by the equations $\vec{r} \cdot (\vec{i} + \vec{j}) = 0$, $\vec{r} \cdot (n\vec{k} + m\vec{j}) = 0$, $\vec{r} \cdot (m\vec{k} + \vec{i}) = 0$ and $\vec{r} \cdot (\vec{i} + m\vec{j} + n\vec{k}) = p$, then the volume of the tetrahedon is:

In the figure, \(\overrightarrow{AE}\) is the vector component of \(\vec{q}\) on \(\vec{p}\). From \(\triangle ABE\), we have \(\overrightarrow{AB} + \overrightarrow{BE} = \overrightarrow{AE}\). If \(\vec{q} + \vec{r} = \dfrac{(\vec{p}\cdot\vec{q})}{(\vec{p}\cdot\vec{q})}\vec{p}\), find \(\vec{r}\):

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=-\hat{i}-8\hat{j}+2\hat{k}$ and $\vec{c}=4\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three vectors. If $\vec{b}\times\vec{a}=\vec{c}\times\vec{a}$, and the angle between the vector $\vec{c}$ and the vector $3\hat{i}+4\hat{j}+\hat{k}$ is $\theta$, then the greatest integer less than or equal to $\tan^2\theta$ is:

If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar, mutually perpendicular unit vectors, then $[\vec{a} \vec{q}] \vec{a} + [\vec{b} \vec{q}] \vec{b} + [\vec{c} \vec{q}] \vec{c}$ is equal to:

Given \(\overrightarrow{OA} = 7\hat{i} - 4\hat{j} + 7\hat{k}\), \(\overrightarrow{OB} = \hat{i} - 6\hat{j} + 10\hat{k}\), \(\overrightarrow{OC} = -\hat{i} - 3\hat{j} + 4\hat{k}\), \(\overrightarrow{OD} = 5\hat{i} - \hat{j} + \hat{k}\). Then \(ABCD\) is

Let a = i + j and b = 2i - k, then the point of intersection of the lines r × a = b × a and r × b = a × b is

A particle has two velocities of equal magnitude inclined to each other at an angle \(\theta\). If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then \(\theta\) is

$p_2$ is equal to:

Let three vectors $\vec{a}=\alpha\hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=5\hat{i}+3\hat{j}+4\hat{k}$, $\vec{c}=x\hat{i}+y\hat{j}+z\hat{k}$ form a triangle such that $\vec{c}=\vec{a}-\vec{b}$ and the area of the triangle is $5\sqrt{6}$. If $\alpha$ is a positive real number, then $|\vec{c}|^2$ is equal to:

Let the position vectors of the vertices $A$, $B$ and $C$ of a tetrahedron $ABCD$ be $\hat{i}+2\hat{j}+\hat{k}$, $\hat{i}+3\hat{j}-2\hat{k}$ and $2\hat{i}+\hat{j}-\hat{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\dfrac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\dfrac{\sqrt{805}}{6\sqrt{2}}$, then the position vector of $E$ is:

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 \(\lambda\).

Let $\overrightarrow{a}=2\hat{i}-\hat{j}+3\hat{k}$, $\overrightarrow{b}=3\hat{i}-5\hat{j}+\hat{k}$ and $\overrightarrow{c}$ be a vector such that $\overrightarrow{a}\times\overrightarrow{c}=\overrightarrow{c}\times\overrightarrow{b}$ and $(\overrightarrow{a}+\overrightarrow{c})\cdot(\overrightarrow{b}+\overrightarrow{c})=168$. Then the maximum value of $|\overrightarrow{c}|^2$ is:

[JEE Main 2020] The volume of a parallelepiped whose coterminous edges are \(\vec{u}=\hat{i}+\hat{j}+\lambda\hat{k}\), \(\vec{v}=\hat{i}+\hat{j}+3\hat{k}\), \(\vec{w}=2\hat{i}+\hat{j}+\hat{k}\) is 1 cubic unit. If \(\theta\) is the angle between the edges \(\vec{u}\) and \(\vec{w}\), then \(\cos\theta\) can be

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}+2\hat{j}+\hat{k}$ and $\vec{d}=\vec{a}\times\vec{b}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot\vec{c}=|\vec{c}|$, $|\vec{c}-2\vec{a}|^2=8$ and the angle between $\vec{d}$ and $\vec{c}$ is $\dfrac{\pi}{4}$, then $|10-3\vec{b}\cdot\vec{c}|+|\vec{d}\times\vec{c}|^2$ is equal to ________.

Let A, B, C be three points in the $xy$-plane, whose position vectors are given by $\sqrt{3}\hat{i}+\hat{j}$, $\hat{i}+\sqrt{3}\hat{j}$ and $a\hat{i}+(1-a)\hat{j}$ respectively with respect to the origin $O$. If the distance of the point $C$ from the line bisecting the angle between the vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is $\dfrac{9}{\sqrt{2}}$, then the sum of all the possible values of $a$ is:

Consider a $\triangle ABC$ where $A(1,2,3)$, $B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle BAC$ meets the line $BC$ at $D$, then the length of the projection of the vector $\overrightarrow{AD}$ on the vector $\overrightarrow{AC}$ is:

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|2\vec{a}+3\vec{b}|=|3\vec{a}+\vec{b}|\). Find the angle between \(\vec{a}\) and \(\vec{b}\).

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 \(6\alpha\).

The vector \(\vec{a} = \alpha\hat{i} + 2\hat{j} + \beta\hat{k}\) lies in the plane of the vectors \(\vec{b} = \hat{i} + \hat{j}\) and \(\vec{c} = \hat{j} + \hat{k}\) and bisects the angle between \(\vec{b}\) and \(\vec{c}\). Then which one of the following gives possible values of \(\alpha\) and \(\beta\)?

Let \(\vec{a}=4\hat{i}+3\hat{j}\) and \(\vec{b}=3\hat{i}-4\hat{j}+5\hat{k}\). If \((\vec{a}+\vec{b})\perp(\lambda\vec{a}-\vec{b})\), find \(\lambda\).

Let the arc $AC$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $AC$ divides the arc $AC$ such that $\dfrac{\text{length of arc }AB}{\text{length of arc }BC}=\dfrac{1}{5}$, and $\overrightarrow{OC}=\alpha\overrightarrow{OA}+\beta\overrightarrow{OB}$, then $\alpha+\sqrt{2}(\sqrt{3}-1)\beta$ is equal to:

Let $\vec{a}=2\hat{i}+5\hat{j}-\hat{k}$, $\vec{b}=2\hat{i}-2\hat{j}+2\hat{k}$ and $\vec{c}$ be three vectors such that $(\vec{c}+\hat{i})\times(\vec{a}+\vec{b}+\hat{i})=\vec{a}\times(\vec{c}+\hat{i})$. If $\vec{a}\cdot\vec{c}=-29$, then $\vec{c}\cdot(-2\hat{i}+\hat{j}+\hat{k})$ is equal to:

Ex. 49 Statement I: If \(\mathbf{a} = 2\mathbf{i} + \mathbf{k}\), \(\mathbf{b} = 3\mathbf{j} + 4\mathbf{k}\) and \(\mathbf{c} = \lambda \mathbf{a} + \mu\mathbf{b}\) are coplanar, then \(\mathbf{c} = 4\mathbf{a} - \mathbf{b}\).Statement II: A set of vectors \(\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3, \ldots, \mathbf{a}_n\) is said to be linearly independent, if every relation of the form \(l_1\mathbf{a}_1 + l_2\mathbf{a}_2 + l_3\mathbf{a}_3 + \cdots + l_n\mathbf{a}_n = 0\) implies that \(l_1 = l_2 = l_3 = \cdots = l_n = 0\) (scalar).

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 \(|\hat{c}\cdot(3\hat{i}-\hat{j}+\hat{k})|\).

If the components of $\vec{a}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$ along and perpendicular to $\vec{b}=3\hat{i}+\hat{j}-\hat{k}$ respectively are $\dfrac{10}{11}(3\hat{i}+\hat{j}-\hat{k})$ and $\dfrac{1}{11}(-4\hat{i}-5\hat{j}-17\hat{k})$, then $\alpha^2+\beta^2+\gamma^2$ is equal to:

Let $\hat{a}$ be a unit vector perpendicular to the vectors $\overrightarrow{b}=\hat{i}-2\hat{j}+3\hat{k}$ and $\overrightarrow{c}=2\hat{i}+3\hat{j}-\hat{k}$, and makes an angle of $\cos^{-1}\!\left(-\dfrac{1}{3}\right)$ with the vector $\hat{i}+\hat{j}+\hat{k}$. If $\hat{a}$ makes an angle of $\dfrac{\pi}{3}$ with the vector $\hat{i}+\alpha\hat{j}+\hat{k}$, then the value of $\alpha$ is:

If $\vec{m}$, $\vec{n}$, $\vec{o}$ are 3 unit vectors such that $\vec{m} \cdot \vec{n} = 0$, $\vec{n} \cdot \vec{o} = 0$ and the angle between $\vec{m}$ and $\vec{o}$ is $\frac{\pi}{6}$, then the value of $|\vec{m} \times \vec{n} - \vec{n} \times \vec{o}|$ is equal to

Let $\vec{a}=3\hat{i}+2\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b})\times\vec{c}=2(\vec{a}\times\vec{b})+24\hat{j}-6\hat{k}$ and $(\vec{a}-\vec{b}+\hat{i})\cdot\vec{c}=-3$. Then $|\vec{c}|^2$ is equal to _____.

Let $\vec{a}=3\hat{i}-\hat{j}+2\hat{k}$, $\vec{b}=\vec{a}\times(\hat{i}-2\hat{k})$ and $\vec{c}=\vec{b}\times\hat{k}$. Then the projection of $\vec{c}-2\hat{j}$ on $\vec{a}$ is:

Let a unit vector which makes an angle of $60^\circ$ with $2\hat{i}+2\hat{j}-\hat{k}$ and angle $45^\circ$ with $\hat{i}-\hat{k}$ be $\overrightarrow{C}$. Then $\overrightarrow{C}+\left(-\dfrac{1}{2}\hat{i}+\dfrac{1}{3\sqrt{2}}\hat{j}-\dfrac{\sqrt{2}}{3}\hat{k}\right)$ is: