Vector Algebra · Mathematics · JEE Advanced

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 $$ 1...

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 $\o...

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 th...

Let $$\overrightarrow u $$, $$\overrightarrow v $$ and $$\overrightarrow w $$ be vectors in three-dimensional space, where $$\overrightarrow u $$ and ...

Let $$\overrightarrow a = 2\widehat i + \widehat j - \widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j + \widehat k$$ be two vectors. Co...

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 + \...

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

Let $$\overrightarrow a \,\,,\,\,\overrightarrow b $$ and $$\overrightarrow c $$ be three non-coplanar unit vectors such that the angle between every ...

If $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ are unit vectors satisfying $${\left| {\overrightarrow a - \overrightarrow b...

Let $$\overrightarrow a = - \widehat i - \widehat k,\overrightarrow b = - \widehat i + \widehat j$$ and $$\overrightarrow c = \widehat i + 2\wide...

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

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}=\fr...

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

|$$\overrightarrow{OX}$$ $$ \times $$ $$\overrightarrow{OY}$$| = ?

Match the following : $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column $$I$$ (A)$$\,\,\,\,$$ In $${R^2},$$ If the magnitude of the projection vector o...

match List $$I$$ with List $$II$$ and select the correct answer using the code given below the lists: $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List $$...

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 ...

If $$\overrightarrow a $$ and $$\overrightarrow b $$ are vectors such that $$\left| {\overrightarrow a + \overrightarrow b } \right| = \sqrt {29} $$...

Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$$ and $$\overrightarrow c...

Match the statements given in Column -$$I$$ with the values given in Column-$$II.$$ $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I$$ (A) $$\,\,\,\,...

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 $...

Two adjacent sides of a parallelogram $$ABCD$$ are given by $$\overrightarrow {AB} = 2\widehat i + 10\widehat j + 11\widehat k$$ and $$\,\overrighta...

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are unit vectors such that $$(\overrightarrow a \times \ov...

The unit vector perpendicular to both $${L_1}$$ and $${L_2}$$ is :

The shortest distance between $${L_1}$$ 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 v...

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $$\overrightarrow a \,,\,\overrightarrow b ,\overrighta...

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

Let $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ be unit vectors such that $${\overrightarrow a + \overrightarrow b + \overrigh...

Let the vectors $$\overrightarrow {PQ} ,\,\,\overrightarrow {QR} ,\,\,\overrightarrow {RS} ,\,\,\overrightarrow {ST} ,\,\,\overrightarrow {TU} ,$$ and...

Let $$\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$$ and $$\overrightarrow ...

If $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ are three non-zero, non-coplanar vectors and $$\overrightarrow {{b_1}} = \overr...

If $$\overrightarrow a = \left( {\widehat i + \widehat j + \widehat k} \right),\overrightarrow a .\overrightarrow b = 1$$ and $$\overrightarrow a \...

The unit vector which is orthogonal to the vector $$3\overrightarrow i + 2\overrightarrow j + 6\overrightarrow k $$ and is coplanar with the vectors...

The value of $$'a'$$ so that the volume of parallelopiped formed by $$\widehat i + a\widehat j + \widehat k,\widehat j + a\widehat k$$ and $$a\widehat...

Let $$\overrightarrow V = 2\overrightarrow i + \overrightarrow j - \overrightarrow k $$ and $$\overrightarrow W = \overrightarrow i + 3\overright...

If $${\overrightarrow a }$$ and $${\overrightarrow b }$$ are two unit vectors such that $${\overrightarrow a + 2\overrightarrow b }$$ and $${5\overri...

Let $$\overrightarrow a = \overrightarrow i - \overrightarrow k ,\overrightarrow b = x\overrightarrow i + \overrightarrow j + \left( {1 - x} \rig...

If $$\overrightarrow a \,,\,\overrightarrow b $$ and $$\overrightarrow c $$ are unit vectors, then $${\left| {\overrightarrow a - \overrightarrow b }...

If the vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ form the sides $$BC,$$ $$CA$$ and $$AB$$ respectively of a triangl...

Let the vectors $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ be such that $$\left( {\overrightarrow a \ti...

If $$\overrightarrow a \,,\,\overrightarrow b $$ and $$\overrightarrow c $$ are unit coplanar vectors, then the scalar triple product $$\left[ {2\over...

Let $$a=2i+j-2k$$ and $$b=i+j.$$ If $$c$$ is a vector such that $$a.$$ $$c = \left| c \right|,\left| {c - a} \right| = 2\sqrt 2 $$ and the angle betw...

Let $$a=2i+j+k, b=i+2j-k$$ and a unit vector $$c$$ be coplanar. If $$c$$ is perpendicular to $$a,$$ then $$c =$$

If $$a = i + j + k,\overrightarrow b = 4i + 3j + 4k$$ and $$c = i + \alpha j + \beta k$$ are linearly dependent vectors and $$\left| c \right| = \sq...

For three vectors $$u,v,w$$ which of the following expression is not equal to any of the remaining three?

Let $$\overrightarrow u ,\overrightarrow v $$ and $$\overrightarrow w $$ be vectors such that $$\overrightarrow u + \overrightarrow v + \overrightar...

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ are non coplanar unit vectors such that $$\overrightarrow a \times \left( {\overright...

Let $$\overrightarrow a = \widehat i - \widehat j,\overrightarrow b = \widehat j - \widehat k,\overrightarrow c = \widehat k - \widehat i.$$ If $$...

If $$\overrightarrow a ,$$ $$\overrightarrow b $$ and $$\overrightarrow c $$ are three non coplanar vectors, then $$\left( {\overrightarrow a + \ov...

Let $$a, b, c$$ be distinct non-negative numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\widehat i + \widehat k$$ and $$c\widehat i...

Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,$$ be three non-coplanar vectors and $$\overrightarrow p ,\overrightarrow q ,\overright...

The number of vectors of unit length perpendicular to vectors $$\overrightarrow a = \left( {1,1,0} \right)$$ and $$\overrightarrow b = \left( {0,1,1...

Let $$\overrightarrow a = {a_1}i + {a_2}j + {a_3}k,\,\,\,\overrightarrow b = {b_1}i + {b_2}j + {b_3}k$$ and $$\overrightarrow c = {c_1}i + {c_2}j +...

For non-zero vectors $${\overrightarrow a ,\,\overrightarrow b ,\overrightarrow c },$$ $$\left| {\left( {\overrightarrow a \times \overrightarrow b }...

The scalar $$\overrightarrow A .\left( {\overrightarrow B + \overrightarrow C } \right) \times \left( {\overrightarrow A + \overrightarrow B + \ove...

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{...

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\...

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 parall...

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...

Let $$\Delta PQR$$ be a triangle. Let $$\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP} $$ and $$\overrightarrow c = \overrightarrow {PQ...

Let $$\overrightarrow x ,\overrightarrow y $$ and $$\overrightarrow z $$ be three vectors each of magnitude $$\sqrt 2 $$ and the angle between each pa...

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 per...

Let $$a$$ and $$b$$ two non-collinear unit vectors. If $$u = a - \left( {a\,.\,b} \right)\,b$$ and $$v = a \times b,$$ then $$\left| v \right|$$ is

Which of the following expressions are meaningful?

The vector $$\,{1 \over 3}\left( {2\widehat i - 2\widehat j + \widehat k} \right)$$ is

Let $$\vec a = 2\hat i - \hat j + \hat k,\vec b = \hat i + 2\hat j - \hat k$$ and $$\overrightarrow c = \widehat i + \widehat j - 2\widehat k - 2\wi...

Match the folowing : (A)$$\,\,\,$$Two rays $$x + y = \left| a \right|$$ and $$ax - y=1$$ intersects each other in the $$\,\,\,\,\,\,\,\,\,\,$$first q...

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 no...

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are distinct vectors such that $$\,\overrightarrow a \tim...

If $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w ,$$ are three non-coplanar unit vectors and $$\alpha ,\beta ,\gamma $$ are the angles bet...

Let $$V$$ be the volume of the parallelopiped formed by the vectors $$\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k,$$ $$...

Find $$3-$$dimensional vectors $${\overrightarrow v _1},{\overrightarrow v _2},{\overrightarrow v _3}$$ satisfying $$\,{\overrightarrow v _1}.{\over...

Let $$\overrightarrow A \left( t \right) = {f_1}\left( t \right)\widehat i + {f_2}\left( t \right)\widehat j$$ and $$$\overrightarrow B \left( t \rig...

Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concur...

Let $$u$$ and $$v$$ be units vectors. If $$w$$ is a vector such that $$w + \left( {w \times u} \right) = v,$$ then prove that $$\left| {\left( {u \tim...

Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points ...

For any two vectors $$u$$ and $$v,$$ prove that (a) $${\left( {u\,.\,v} \right)^2} + {\left| {u \times v} \right|^2} = {\left| u \right|^2}{\left| v ...

If $$A,B$$ and $$C$$ are vectors such that $$\left| B \right| = \left| C \right|.$$ Prove that $$\left[ {\left( {A + B} \right) \times \left( {A + C}...

If the vectors $$\overrightarrow b ,\overrightarrow c ,\overrightarrow d ,$$ are not coplanar, then prove that the vector $$\left( {\overrightarrow a...

In a triangle $$ABC, D$$ and $$E$$ are points on $$BC$$ and $$AC$$ respectively, such that $$BD=2DC$$ and $$AE=3EC.$$ Let $$P$$ be the point of inters...

Determine the value of $$'c'$$ so that for all real $$x,$$ the vector $$cx\widehat i - 6\widehat j - 3\widehat k$$ and $$x\widehat i + 2\widehat j + ...

Let $$\overrightarrow A = 2\overrightarrow i + \overrightarrow k ,\,\overrightarrow B = \overrightarrow i + \overrightarrow j + \overrightarrow k...

If vectors $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$ are coplanar, show that $$$\left| {\matrix{ {} & {\overrightarrow {a.}...

In a triangle $$OAB,E$$ is the midpoint of $$BO$$ and $$D$$ is a point on $$AB$$ such that $$AD:DB=2:1.$$ If $$OD$$ and $$AE$$ intersect at $$P,$$ det...

Let $$OA$$ $$CB$$ be a parallelogram with $$O$$ at the origin and $$OC$$ a diagonal. Let $$D$$ be the midpoint of $$OA.$$ Using vector methods prove ...

If $$A, B, C, D$$ are any four points in space, prove that - $$\left| {\overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \ti...

The position vectors of the points $$A, B, C$$ and $$D$$ are $$3\widehat i - 2\widehat j - \widehat k,\,2\widehat i + 3\widehat j - 4\widehat k,\, - \...

$${A_1},{A_2},.................{A_n}$$ are the vertices of a regular plane polygon with $$n$$ sides and $$O$$ is its centre. Show that $$\sum\limits_...

Find all values of $$\lambda $$ such that $$x, y, z,$$$$\, \ne $$$$(0,0,0)$$ and $$\left( {\overrightarrow i + \overrightarrow j + 3\overrightarrow...

Let $$OA=a,$$ $$OB=10a+2b$$ and $$OC=b$$ where $$O,A$$ and $$C$$ are non-collinear points. Let $$p$$ denote the area of the quadrilateral $$OABC,$$ an...

A nonzero vector $$\overrightarrow a $$ is parallel to the line of intersection of the plane determined by the vectors $$\widehat i,\widehat i + \wide...

If $$\overrightarrow b \,$$ and $$\overrightarrow c \,$$ are two non-collinear unit vectors and $$\overrightarrow a \,$$ is any vector, then $$\left( ...

A unit vector coplanar with $$\overrightarrow i + \overrightarrow j + 2\overrightarrow k $$ and $$\overrightarrow i + 2\overrightarrow j + \overri...

Given that $$\overrightarrow a = \left( {1,1,1} \right),\,\,\overrightarrow c = \left( {0,1, - 1} \right),\,\overrightarrow a .\overrightarrow b = ...

The components of a vector $$\overrightarrow a $$ along and perpendicular to a non-zero vector $$\overrightarrow b $$ are ......and .....respectively.

If the vectors $$a\widehat i + \widehat j + \widehat k,\,\,\widehat i + b\widehat j + \widehat k$$ and $$\widehat i + \widehat j + c\widehat k$$ $$\...

Let $$b = 4\widehat i + 3\widehat j$$ and $$\overrightarrow c $$ be two vectors perpendicular to each other in the $$xy$$-plane. All vectors in the sa...

If $$\overrightarrow A \overrightarrow {\,B} \overrightarrow {\,C} $$ are three non-coplannar vectors, then - $${{\overrightarrow A .\overrightarrow ...

If $$\overrightarrow A = \left( {1,1,1} \right),\,\,\overrightarrow C = \left( {0,1, - 1} \right)$$ are given vectors, then a vector $$B$$ satifying...

$$A, B, C$$ and $$D,$$ are four points in a plane with position vectors $$a, b, c$$ and $$d$$ respectively such that $$$\left( {\overrightarrow a - \...

Let $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$ be vectors of length $$3, 4, 5$$ respectively. Let $$\overrightarrow A $$ be perpendi...

For any three vectors $${\overrightarrow a ,\,\overrightarrow b ,}$$ and $${\overrightarrow c ,}$$ $$\left( {\overrightarrow a - \overrightarrow b }...

The points with position vectors $$a+b,$$ $$a-b,$$ and $$a+kb$$ are collinear for all real values of $$k.$$

If $$X.A=0, X.B=0, X.C=0$$ for some non-zero vector $$X,$$ then $$\left[ {A\,B\,C} \right] = 0$$

Let $$\overrightarrow A ,\overrightarrow B $$ and $${\overrightarrow C }$$ be unit vectors suppose that $$\overrightarrow A .\overrightarrow B = \ove...