Algebra Questions (626)

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

The final resultant displacement of the ant is:

Direction of the ant's resultant displacement is :

Ex. 109: If AP, BQ and CR are the altitudes of acute \triangle ABC and 9\overrightarrow{AP} + 4\overrightarrow{BQ} + 7\overrightarrow{CR} = \mathbf{0}, then \angle ACB is equal to

Example 29. The position vectors of the vertices A, B and C of a triangle are \(\vec{i} - \vec{j} - 3\vec{k}\), \(2\vec{i} + \vec{j} - 2\vec{k}\) and \(-5\vec{i} + 2\vec{j} - 6\vec{k}\), respectively. The length of the bisector AD of the \(\angle BAC\), where D is on the segment BC, is

If \(\vec{a} = \dfrac{1}{\sqrt{10}}(3\hat{i} + \hat{k})\) and \(\vec{b} = \dfrac{1}{7}(2\hat{i} + 3\hat{j} - 6\hat{k})\), then the value of \((2\vec{a} - \vec{b}) \cdot [(\vec{a} \times \vec{b}) \times (\vec{a} + 2\vec{b})]\) is

If AB AC AB × AC = λ AE AG AE × AG, then the value of λ is:

The volume of the parallelepiped whose coterminous edges are represented by the vectors $2\mathbf{b} \times \mathbf{c}$, $3\mathbf{c} \times \mathbf{a}$ and $4\mathbf{a} \times \mathbf{b}$ where $\mathbf{a} = (1 + \sin \theta)\mathbf{i} + \cos \theta \mathbf{j} + \sin 2\theta \mathbf{k}$, $\mathbf{b} = \sin\left(\theta + \frac{2\pi}{3}\right)\mathbf{i} + \cos\left(\theta + \frac{2\pi}{3}\right)\mathbf{j} + \sin\left(2\theta + \frac{4\pi}{3}\right)\mathbf{k}$, $\mathbf{c} = \sin\left(\theta - \frac{2\pi}{3}\right)\mathbf{i} + \cos\left(\theta - \frac{2\pi}{3}\right)\mathbf{j} + \sin\left(2\theta - \frac{4\pi}{3}\right)\mathbf{k}$ is 18 cubic units, then the value of $\theta$ in the interval $\left(0, \frac{\pi}{2}\right)$ is:

Ex. 22 If a, b, c be non-zero vectors such that a is perpendicular to b and c and \(|\mathbf{a}| = 1\), \(|\mathbf{b}| = 2\), \(|\mathbf{c}| = 1\), \(\mathbf{b} \cdot \mathbf{c} = 1\) and there is a non-zero vector d coplanar with \(\mathbf{a} + \mathbf{b}\) and \(2\mathbf{b} - \mathbf{c}\) and \(\mathbf{d} \cdot \mathbf{a} = 1\), then minimum value of \(|\mathbf{d}|\) is

If a, b and c are three non-coplanar uni-modular vectors, each inclined with other at an angle 30°, then volume of tetrahedron whose edges are a, b and c is

If [a × b b × c c × a] = λ[a b c]², then λ is equal to

Let \(\vec{a}=\hat{i}-\hat{j}\), \(\vec{b}=\hat{i}+\hat{j}+\hat{k}\) and \(\vec{c}\) be a vector such that \(\vec{a}\times\vec{c}+\vec{b}=\vec{0}\) and \(\vec{a}\cdot\vec{c}=4\), then \(|\vec{c}|^2\) is equal to:

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha\hat{i}-2\hat{j}+2\hat{k}$ and $\alpha\hat{i}+2\alpha\hat{j}-2\hat{k}$ is acute, is ___.

Let $\vec{a}=3\hat{i}+\hat{j}-2\hat{k}$, $\vec{b}=4\hat{i}+\hat{j}+7\hat{k}$ and $\vec{c}=\hat{i}-3\hat{j}+4\hat{k}$. If a vector $\vec{p}$ satisfies $\vec{p}\times\vec{b}=\vec{c}\times\vec{b}$ and $\vec{p}\cdot\vec{a}=0$, then $\vec{p}\cdot(\hat{i}-\hat{j}-\hat{k})$ is equal to:

If three points A, B and C have position vectors \((1, x, 3)\), \((3, 4, 7)\) and \((y, -2, -5)\) respectively and if they are collinear, then \((x, y)\) is equal to

Consider three vectors $\vec{a},\vec{b},\vec{c}$. Let $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{a}=\vec{b}\times\vec{c}$. If $\alpha\in\left[0,\dfrac{\pi}{3}\right]$ is the angle between the vectors $\vec{b}$ and $\vec{c}$, then the minimum value of $27|\vec{c}-\vec{a}|^2$ is equal to:

Let position vectors of $A,B,C,D$ be $5\hat{i}+5\hat{j}+2\lambda\hat{k}$, $\hat{i}+2\hat{j}+3\hat{k}$, $-2\hat{i}+\lambda\hat{j}+4\hat{k}$ and $-\hat{i}+5\hat{j}+6\hat{k}$. Let $S=\{\lambda\in\mathbb{R}: A,B,C,D\text{ coplanar}\}$. Then $\sum_{\lambda\in S}(\lambda+2)^2$ is equal to