Vector Algebra Questions (573)

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:

Let $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$, $\vec{b}=3\hat{i}+4\hat{j}-5\hat{k}$ and a vector $\vec{c}$ be such that $\vec{a}\times(\vec{b}+\vec{c})+\vec{b}\times\vec{c}=\hat{i}+8\hat{j}+13\hat{k}$. If $\vec{a}\cdot\vec{c}=13$, then $(24-\vec{b}\cdot\vec{c})$ is equal to _____

Let $\overrightarrow{OA}=\vec{a}$, $\overrightarrow{OB}=12\vec{a}+4\vec{b}$ and $\overrightarrow{OC}=\vec{b}$, where $O$ is the origin. If $S$ is the parallelogram with adjacent sides $OA$ and $OC$, then $\dfrac{\text{Area of quadrilateral }OABC}{\text{Area of }S}$ is equal to:

Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=3(\hat{i}-\hat{j}+\hat{k})$. Let $\vec{c}$ be a vector such that $\vec{a}\times\vec{c}=\vec{b}$ and $\vec{a}\cdot\vec{c}=3$. Then $\vec{a}\cdot\left[(\vec{c}\times\vec{b})-\vec{b}-\vec{c}\right]$ is equal to:

Let $\vec{a}=9\hat{i}-13\hat{j}+25\hat{k}$, $\vec{b}=3\hat{i}+7\hat{j}-13\hat{k}$ and $\vec{c}=17\hat{i}-2\hat{j}+\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times\vec{a}=(\vec{b}+\vec{c})\times\vec{a}$ and $\vec{r}\cdot(\vec{b}-\vec{c})=0$, then $\dfrac{|593\vec{r}+67\vec{a}|^2}{(593)^2}$ is equal to _____

ABC is a triangle, right angled at A. The resultant of the forces acting along \(\overrightarrow{AB}\), \(\overrightarrow{AC}\) with magnitudes \(1/AB\) and \(1/AC\) respectively is the force along \(\overrightarrow{AD}\), where D is the foot of the perpendicular from A onto BC. The magnitude of the resultant is

Let the position vectors of \(A\), \(B\), \(C\) and \(D\) be \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) and \(\vec{d}\) respectively. Given that \(OA : CB = 2 : 1\) and \(OD : AB = 1 : 3\), and \(OX : XC = \lambda : 1\), \(AX : XD = \mu : 1\). Find \(OX : XC\).

Let a unit vector $\hat{u}=x\hat{i}+y\hat{j}+z\hat{k}$ make angles $\dfrac{\pi}{2}$, $\dfrac{\pi}{3}$ and $\dfrac{2\pi}{3}$ with the vectors $\dfrac{1}{\sqrt{2}}\hat{i}+\dfrac{1}{\sqrt{2}}\hat{k}$, $\dfrac{1}{\sqrt{2}}\hat{j}+\dfrac{1}{\sqrt{2}}\hat{k}$ and $\dfrac{1}{\sqrt{2}}\hat{i}+\dfrac{1}{\sqrt{2}}\hat{j}$ respectively. If $\vec{v}=\dfrac{1}{\sqrt{2}}\hat{i}+\dfrac{1}{\sqrt{2}}\hat{j}+\dfrac{1}{\sqrt{2}}\hat{k}$, then $|\hat{u}-\vec{v}|^2$ is equal to:

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=3\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a}\cdot\vec{c}=5$, then $|\vec{c}|$ is equal to:

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

[JEE Main 2021] If a unit vector \(\hat{a}\) makes angles \(\dfrac\pi3\) with \(\hat{i}\), \(\dfrac\pi4\) with \(\hat{j}\) and an acute angle \(\theta\) with \(\hat{k}\), then \(\theta\) equals

Vectors \(\vec{a}\) and \(\vec{b}\) make an angle \(\theta=\frac{2\pi}{3}\). If \(|\vec{a}|=1,\;|\vec{b}|=2\), then the minimum value of \(|2\vec{a}+\vec{b}|^2+|2\vec{a}-\vec{b}|^2\) over all configurations is

84. Given parallelopiped formed by the vectors \(\hat{i} + \lambda\hat{j} + \hat{k}\), \(\hat{j} + \lambda\hat{k}\) and \(\lambda\hat{i} + \hat{k}\). Find the minimum volume of the parallelopiped (rounded to 3 decimal places).

[JEE Main 2022] Let \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\), \(\vec{b}=\hat{i}-\hat{j}+2\hat{k}\), \(\vec{c}=5\hat{i}+3\hat{j}-\hat{k}\). If \(\alpha\) is the projection of \((\vec{a}+\vec{b})\) on \(\vec{c}\), and \(\beta\) is the projection of \(\vec{c}\) on \((\vec{a}+\vec{b})\), find \(6(\alpha+\beta)\).

Given \(\vec{a} = \hat{i} + 2\hat{j} + 4\hat{k}\), \(\vec{b} = \hat{i} + \lambda \hat{j} + 4\hat{k}\) and \(\vec{c} = 2\hat{i} + 4\hat{j} + (\lambda^2 - 1)\hat{k}\) are coplanar vectors. Find \(\vec{a} \times \vec{c}\).

The value of \(|(p + q)\cos\theta + r|\) is

The value of \begin{vmatrix} a \cdot p & b \cdot p & c \cdot p \\ a \cdot q & b \cdot q & c \cdot q \\ \cdot & \cdot & \cdot \end{vmatrix} is

Let ABCD be a parallelogram such that AB = q, AD = p and ∠BAD be an acute angle. If r is the vector that coincides with the altitude directed from the vertex B to the side AD, then r is given by

Let \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) be three unit vectors such that \(\vec{a}\times(\vec{b}\times\vec{c})=\dfrac{\sqrt{3}}{2}(\vec{b}+\vec{c})\). If \(\vec{b}\) is not parallel to \(\vec{c}\), then the angle between \(\vec{a}\) and \(\vec{b}\) is

Let \(\vec{b} = x\hat{i} + y\hat{j} + z\hat{k}\). It is given that \(\vec{a} \times \vec{b} + \vec{c} = 0\), \(\vec{c} = -\vec{a} \times \vec{b} = \vec{b} \times \vec{a}\), and \(\vec{c}\) is normal to \(\vec{b}\). Also \(\vec{a} \cdot \vec{b} = 3\). Then \(\vec{b}\) is:

P is the point of intersection of the diagonals of the parallelogram ABCD. If O is any point, then $\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD}$ is equal to

Ex. 96 Statement I: If a is perpendicular to b and c, then a × (b × c) = 0Statement II: If b is perpendicular to c, then b × c = 0

We have \((2\vec{a} - \vec{b}) \cdot [(\vec{a} \times \vec{b}) \times (\vec{a} + 2\vec{b})]\). If \(|\vec{a}| = 1\), \(|\vec{b}| = 1\), \(\vec{a} \cdot \vec{b} = 1\), find the value of the expression.

Ex. 53 In parallelogram ABCD, M is a point on DC which divides DC in the ratio 1 : 2 and AM intersects BD at Q. Point Q divides DB in the ratio