Vector Algebra

Let $ \vec{a}, \vec{b} $ be two vectors, and let P, Q and R be the points with position vectors $ \vec{a}, \vec{b} $ and $ \vec{a} + \vec{b} $, respectively, with respect to the origin O. If $ |\vec{a} + \vec{b}| = \sqrt{21} $, $ |\vec{a} -...

For real numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and $\mu$, consider the matrix
$$ M = \begin{bmatrix} \alpha & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \beta & \frac{1}{\sqrt{3}} \\ \gamma & \delta & \mu \end{...

Consider the vectors
\(\vec{x}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k}, \quad \vec{y}=2 \hat{\imath}+3 \hat{\jmath}+\hat{k}, \quad \text { and } \quad \vec{z}=3 \hat{\imath}+\hat{\jmath}+2 \hat{k}\)
For two distinct positive real numbers $\...

Let $\vec{w} = \hat{i} + \hat{j} - 2\hat{k}$, and $\vec{u}$ and $\vec{v}$ be two vectors, such that $\vec{u} \times \vec{v} = \vec{w}$ and $\vec{v} \times \vec{w} = \vec{u}$. Let $\alpha, \beta, \gamma$, and $t$ be real numbers such that
$...

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\vec{0}$ denote the zero vector. Let $P, Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside th...

Let $\vec{p}=2 \hat{i}+\hat{j}+3 \hat{k}$ and $\vec{q}=\hat{i}-\hat{j}+\hat{k}$. If for some real numbers $\alpha, \beta$, and $\gamma$, we have

$$ 15 \hat{i}+10 \hat{j}+6 \hat{k}=\alpha(2 \vec{p}+\vec{q})+\beta(\vec{p}-2 \vec{q})+\gamma(\...

Let $\overrightarrow{O P}=\frac{\alpha-1}{\alpha} \hat{i}+\hat{j}+\hat{k}, \overrightarrow{O Q}=\hat{i}+\frac{\beta-1}{\beta} \hat{j}+\hat{k}$ and $\overrightarrow{O R}=\hat{i}+\hat{j}+\frac{1}{2} \hat{k}$ be three vectors, where $\alpha, \...

Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}$, $\vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}$ and $\vec{d}=2 \hat{i}+\hat{j}+\hat{k}$,...

Let $P$ be the plane $\sqrt{3} x+2 y+3 z=16$ and let
$S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}: \alpha^2+\beta^2+\gamma^2=1\right.$ and the distance of $(\alpha, \beta, \gamma)$ from the plane $P$ is $\left.\frac{7}{2}\right\}$....

Let $\hat{\imath}, \hat{\jmath}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let

$$
\begin{aligned}
& \vec{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k} \text {, } \\
& \vec{b}=\hat{\imath}+b_{2} \hat{\jmath}+b_{3}...

Let O be the origin and \(\overrightarrow {OA} = 2\widehat i + 2\widehat j + \widehat k\) and \(\overrightarrow {OB} = \widehat i - 2\widehat j + 2\widehat k\) and $$\overrightarrow {OC} = {1 \over 2}\left( {\overrightarrow {OB} - \lamb...

Let \(\overrightarrow u\), \(\overrightarrow v\) and \(\overrightarrow w\) be vectors in three-dimensional space, where \(\overrightarrow u\) and \(\overrightarrow v\) are unit vectors which are not perpendicular to each other and $$\o...

Let a and b be positive real numbers. Suppose \(PQ = a\widehat i + b\widehat j\) and \(PS = a\widehat i - b\widehat j\) are adjacent sides of a parallelogram PQRS. Let u and v be the projection vectors of \(w = \widehat i + \widehat j\) alo...

Let \(\overrightarrow a = 2\widehat i + \widehat j - \widehat k\) and \(\overrightarrow b = \widehat i + 2\widehat j + \widehat k\) be two vectors. Consider a vector c = \(\alpha\)\(\overrightarrow a\) + \(\beta\)\(\overrightarrow b\), $$...

Let a and b be two unit vectors such that a . b = 0. For some x, y\(\in\)R, let \(\overrightarrow c = x\overrightarrow a + y\overrightarrow b + \overrightarrow a \times \overrightarrow b\). If | \(\overrightarrow c\)| = 2 and the ve...

Let O be the origin and let PQR be an arbitrary triangle. The point S is such that\(\overrightarrow{OP}\) . \(\overrightarrow{OQ}\) + \(\overrightarrow{OR}\) . \(\overrightarrow{OS}\) = \(\overrightarrow{OR}\) . \(\overrightarrow{OP}\) + $$...

|\(\overrightarrow{OX}\) \(\times\) \(\overrightarrow{OY}\)| = ?

Let \(\widehat u = {u_1} \widehat i + {u_2}\widehat j + {u_3}\widehat k\) be a unit vector in \({{R^3}}\) and \(\widehat w = {1 \over {\sqrt 6 }}\left( {\widehat i + \widehat j + 2\widehat k} \right).\) Given that there exists a vector $${...

Suppose that \(\overrightarrow p ,\overrightarrow q\) and \(\overrightarrow r\) are three non-coplanar vectors in \({R^3}\). Let the components of a vector \(\overrightarrow s\) along \(\overrightarrow p ,\) \(\overrightarrow q\) and $$...

Match the following :




Column I

Column II







(A)

In $ \mathbb{R}^2 $, if the magnitude of the projection vector of the vector

$ \alpha \hat{i} + \beta \hat{j} $ on

$ \sqrt{3}\hat{i} + \h...

Let \(\Delta PQR\) be a triangle. Let \(\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP}\) and \(\overrightarrow c = \overrightarrow {PQ} .\) If $$\left| {\overrightarrow a } \right| = 12,\,\,\left| {\overrightarrow b } \right...

Let \(\overrightarrow a \,\,,\,\,\overrightarrow b\) and \(\overrightarrow c\) be three non-coplanar unit vectors such that the angle between every pair of them is \({\pi \over 3}.\) If $$\overrightarrow a \times \overrightarrow b + \o...

Let \(\overrightarrow x ,\overrightarrow y\) and \(\overrightarrow z\) be three vectors each of magnitude \(\sqrt 2\) and the angle between each pair of them is \({\pi \over 3}\). If \(\overrightarrow a\) is a non-zero vector perpendic...

match List \(I\) with List \(II\) and select the correct answer using the code given below the lists:
\(\,\,\,\,\) \(\,\,\,\,\) \(\,\,\,\,\) List \(I\)
(P.)\(\,\,\,\,\) Volume of parallelopiped determined by vectors $$\overrightarrow a ,\...

Let $\overrightarrow{\mathrm{PR}}=3 \hat{i}+\hat{j}-2 \hat{k}$ and $ \overrightarrow{\mathrm{SQ}}=\hat{i}-3 \hat{j}-4 \hat{k}$ determine diagonals of a parallelogram $P Q R S$ and $\overrightarrow{\mathrm{PT}}=\hat{i}+2 \hat{j}+3 \hat{k}$ b...

If \(\overrightarrow a\) and \(\overrightarrow b\) are vectors such that \(\left| {\overrightarrow a + \overrightarrow b } \right| = \sqrt {29}\) and $$\,\overrightarrow a \times \left( {2\widehat i + 3\widehat j + 4\widehat k} \right...

If \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) are unit vectors satisfying
$${\left| {\overrightarrow a - \overrightarrow b } \right|^2} + {\left| {\overrightarrow b - \overrightarrow c } \right|^2} + {\left| {\o...

Match the statements given in Column -\(I\) with the values given in Column-\(II.\)
\(\,\,\,\,\) \(\,\,\,\,\) \(\,\,\,\,\) Column-\(I\)
(A) \(\,\,\,\,\)If $$\overrightarrow a = \widehat j + \sqrt 3 \widehat k,\overrightarrow b = - \wideh...

Let \(\overrightarrow a = - \widehat i - \widehat k,\overrightarrow b = - \widehat i + \widehat j\) and \(\overrightarrow c = \widehat i + 2\widehat j + 3\widehat k\) be three given vectors. If \(\overrightarrow r\) is a vector such t...

Let \(\overrightarrow a = \widehat i + \widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k\) and \(\overrightarrow c = \widehat i - \widehat j - \widehat k\) be three vectors. A vector $$\overrightarrow v ...

The vector (s) which is/are coplanar with vectors \({\widehat i + \widehat j + 2\widehat k}\) and \({\widehat i + 2\widehat j + \widehat k,}\) and perpendicular to the vector \({\widehat i + \widehat j + \widehat k}\) is/are

Two adjacent sides of a parallelogram \(ABCD\) are given by
\(\overrightarrow {AB} = 2\widehat i + 10\widehat j + 11\widehat k\) and \(\,\overrightarrow {AD} = -\widehat i + 2\widehat j + 2\widehat k\)
The side \(AD\) is rotated by an a...

If \(\overrightarrow a\) and \(\overrightarrow b\) are vectors in space given by \(\overrightarrow a = {{\widehat i - 2\widehat j} \over {\sqrt 5 }}\) and $$\overrightarrow b = {{2\widehat i + \widehat j + 3\widehat k} \over {\sqrt {14}...

Let \(P,Q,R\) and \(S\) be the points on the plane with position vectors \({ - 2\widehat i - \widehat j,4\widehat i,3\widehat i + 3\widehat j}\) and \({ - 3\widehat i + 2\widehat j}\) respectively. The quadrilateral \(PQRS\) must be a

If \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) and \(\overrightarrow d\) are unit vectors such that \((\overrightarrow a \times \overrightarrow b )\,.\,(\overrightarrow c \times \overrightarrow d ) = 1\) and $$\overright...

The shortest distance between \({L_1}\) and \({L_2}\) is :

The unit vector perpendicular to both \({L_1}\) and \({L_2}\) is :

Let two non-collinear unit vectors \(\widehat a\) and \(\widehat b\) form an acute angle. A point \(P\) moves so that at any time \(t\) the position vector \(\overrightarrow {OP}\) (where \(O\) is the origin) is given by $$\widehat a\cos t...

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors \(\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c\) such that $$\widehat a\,.\,\widehat b = \widehat b\,.\,\widehat c = \widehat c\,...

Let \(\vec{a}, \vec{b}, \vec{c}\) be unit vectors such that \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\). Which one of the following is correct?

Let the vector \(\overrightarrow {PQ} ,\overrightarrow {QR} ,\overrightarrow {RS} ,\overrightarrow {ST} ,\overrightarrow {TU}\) and \(\overrightarrow {UP}\), represent the sides of a regular hexagon.
Statement 1 : $$\overrightarrow {PQ} ...

The number of distinct real values of \(\lambda\), for which the vectors \(- {\lambda ^2}\widehat i + \widehat j + \widehat k,\widehat i - {\lambda ^2}\widehat j + \widehat k\) and \(\widehat i + \widehat j - {\lambda ^2}\widehat k\) are c...

The minimum of distinct real values of \(\lambda ,\) for which the vectors \(- {\lambda ^2}\widehat i + \widehat j + \widehat k,\) \(\widehat i - {\lambda ^2}\widehat j + \widehat k\) and $$\widehat i + \widehat j - {\lambda ^2}\widehat k$...

Let \(\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c\) be unit vectors such that \({\overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow 0 .}\) Which one of the following is correct ?

Let the vectors \(\overrightarrow {PQ} ,\,\,\overrightarrow {QR} ,\,\,\overrightarrow {RS} ,\,\,\overrightarrow {ST} ,\,\,\overrightarrow {TU} ,\) and \(\overrightarrow {UP} ,\) represent the sides of a regular hexagon.
STATEMENT-1: $$\over...

.tg {border-collapse:collapse;border-spacing:0;}

.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;

overflow:hidden;padding:10px 5px;word-break:normal;}

.tg th{border-color:bl...

Let \(\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k\) and \(\overrightarrow c = \widehat i + \widehat j - \widehat k.\) A vector in the plane of $$\overrightarrow a ...

If \(\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c\) are three non-zero, non-coplanar vectors and
$$\overrightarrow {{b_1}} = \overrightarrow b - {{\overrightarrow b .\,\overrightarrow a } \over {{{\left| {\overrightarrow...

Incident ray is along the unit vector \(\hat{v}\) and the reflected ray is along the unit vector \(\widehat{w}\). The normal is along unit vector \(\hat{a}\) outwards. Express \(\hat{w}\), in terms of \(\hat{a}\) and \(\hat{v}\).

If the incident ray on a surface is along the unit vector \(\widehat v\,\,,\) the reflected ray is along the unit vector \(\widehat w\,\,\) and the normal is along unit vector \(\widehat a\,\,\) outwards. Express \(\widehat w\,\,\) in term...