# Vector Algebra · Mathematics · JEE Advanced

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## Numerical

JEE Advanced 2024 Paper 2 Online
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 $$1... JEE Advanced 2024 Paper 1 Online Let \overrightarrow{O P}=\frac{\alpha-1}{\alpha} \hat{i}+\hat{j}+\hat{k}, \overrightarrow{O Q}=\hat{i}+\frac{\beta-1}{\beta} \hat{j}+\hat{k} and \o... JEE Advanced 2023 Paper 1 Online Let P be the plane \sqrt{3} x+2 y+3 z=16 and let S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}: \alpha^2+\beta^2+\gamma^2=1\right. and th... JEE Advanced 2021 Paper 1 Online Let$$\overrightarrow u $$,$$\overrightarrow v $$and$$\overrightarrow w $$be vectors in three-dimensional space, where$$\overrightarrow u $$and ... JEE Advanced 2019 Paper 2 Offline Let$$\overrightarrow a = 2\widehat i + \widehat j - \widehat k$$and$$\overrightarrow b = \widehat i + 2\widehat j + \widehat k$$be two vectors. Co... JEE Advanced 2018 Paper 1 Offline Let a and b be two unit vectors such that a . b = 0. For some x, y$$ \in $$R, let$$\overrightarrow c = x\overrightarrow a + y\overrightarrow b + \...
JEE Advanced 2015 Paper 2 Offline
Suppose that $$\overrightarrow p ,\overrightarrow q$$ and $$\overrightarrow r$$ are three non-coplanar vectors in $${R^3}$$. Let the components of a...
JEE Advanced 2014 Paper 1 Offline
Let $$\overrightarrow a \,\,,\,\,\overrightarrow b$$ and $$\overrightarrow c$$ be three non-coplanar unit vectors such that the angle between every ...
JEE Advanced 2013 Paper 1 Offline
Consider the set of eight vectors $$V = \left\{ {a\widehat i + b\widehat j + c\widehat k:a,b.c \in \left\{ { - 1,1} \right\}} \right\}.$$ Three non-c...
IIT-JEE 2012 Paper 1 Offline
If $$\overrightarrow a ,\overrightarrow b$$ and $$\overrightarrow c$$ are unit vectors satisfying $${\left| {\overrightarrow a - \overrightarrow b... IIT-JEE 2011 Paper 2 Offline Let$$\overrightarrow a = - \widehat i - \widehat k,\overrightarrow b = - \widehat i + \widehat j$$and$$\overrightarrow c = \widehat i + 2\wide...
IIT-JEE 2010 Paper 1 Offline
If $$\overrightarrow a$$ and $$\overrightarrow b$$ are vectors in space given by $$\overrightarrow a = {{\widehat i - 2\widehat j} \over {\sqrt 5 }... ## MCQ (Single Correct Answer) JEE Advanced 2023 Paper 2 Online Let the position vectors of the points P, Q, R and S be \vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}, \vec{c}=\fr... JEE Advanced 2017 Paper 2 Offline Let O be the origin and let PQR be an arbitrary triangle. The point S is such that$$\overrightarrow{OP}$$.$$\overrightarrow{OQ}$$+$$\overrightarro...
JEE Advanced 2017 Paper 2 Offline
|$$\overrightarrow{OX}$$ $$\times$$ $$\overrightarrow{OY}$$| = ?
JEE Advanced 2015 Paper 1 Offline
Match the following : $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column $$I$$ (A)$$\,\,\,\,$$ In $${R^2},$$ If the magnitude of the projection vector o...
JEE Advanced 2013 Paper 2 Offline
match List $$I$$ with List $$II$$ and select the correct answer using the code given below the lists: $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List $$... IIT-JEE 2012 Paper 2 Offline If$$\overrightarrow a $$and$$\overrightarrow b $$are vectors such that$$\left| {\overrightarrow a + \overrightarrow b } \right| = \sqrt {29} $$... IIT-JEE 2011 Paper 1 Offline Let$$\overrightarrow a = \widehat i + \widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$$and$$\overrightarrow c...
IIT-JEE 2011 Paper 2 Offline
Match the statements given in Column -$$I$$ with the values given in Column-$$II.$$ $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I$$ (A) $$\,\,\,\,... IIT-JEE 2010 Paper 1 Offline Let$$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}$$and ... IIT-JEE 2010 Paper 2 Offline Two adjacent sides of a parallelogram$$ABCD$$are given by$$\overrightarrow {AB} = 2\widehat i + 10\widehat j + 11\widehat k$$and$$\,\overrighta...
IIT-JEE 2009 Paper 1 Offline
If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ and $$\overrightarrow d$$ are unit vectors such that $$(\overrightarrow a \times \ov... IIT-JEE 2008 Paper 2 Offline Let two non-collinear unit vectors$$\widehat a$$and$$\widehat b$$form an acute angle. A point$$P$$moves so that at any time$$t$$the position v... IIT-JEE 2008 Paper 2 Offline The unit vector perpendicular to both$${L_1}$$and$${L_2}$$is : IIT-JEE 2008 Paper 2 Offline The shortest distance between$${L_1}$$and$${L_2}$$is : IIT-JEE 2008 Paper 1 Offline The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors$$\overrightarrow a \,,\,\overrightarrow b ,\overrighta...
IIT-JEE 2007
The minimum of distinct real values of $$\lambda ,$$ for which the vectors $$- {\lambda ^2}\widehat i + \widehat j + \widehat k,$$ $$\widehat i - {\l... IIT-JEE 2007 Let$$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$be unit vectors such that$${\overrightarrow a + \overrightarrow b + \overrigh...
IIT-JEE 2007
Let the vectors $$\overrightarrow {PQ} ,\,\,\overrightarrow {QR} ,\,\,\overrightarrow {RS} ,\,\,\overrightarrow {ST} ,\,\,\overrightarrow {TU} ,$$ and...
IIT-JEE 2006
Let $$\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$$ and $$\overrightarrow ... IIT-JEE 2005 Screening If$$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$are three non-zero, non-coplanar vectors and$$\overrightarrow {{b_1}} = \overr...
IIT-JEE 2004 Screening
If $$\overrightarrow a = \left( {\widehat i + \widehat j + \widehat k} \right),\overrightarrow a .\overrightarrow b = 1$$ and $$\overrightarrow a \... IIT-JEE 2004 Screening The unit vector which is orthogonal to the vector$$3\overrightarrow i + 2\overrightarrow j + 6\overrightarrow k $$and is coplanar with the vectors... IIT-JEE 2003 Screening The value of$$'a'$$so that the volume of parallelopiped formed by$$\widehat i + a\widehat j + \widehat k,\widehat j + a\widehat k$$and$$a\widehat...
IIT-JEE 2002 Screening
If $${\overrightarrow a }$$ and $${\overrightarrow b }$$ are two unit vectors such that $${\overrightarrow a + 2\overrightarrow b }$$ and $${5\overri... IIT-JEE 2002 Screening Let$$\overrightarrow V = 2\overrightarrow i + \overrightarrow j - \overrightarrow k $$and$$\overrightarrow W = \overrightarrow i + 3\overright...
IIT-JEE 2001 Screening
Let $$\overrightarrow a = \overrightarrow i - \overrightarrow k ,\overrightarrow b = x\overrightarrow i + \overrightarrow j + \left( {1 - x} \rig... IIT-JEE 2001 Screening If$$\overrightarrow a \,,\,\overrightarrow b $$and$$\overrightarrow c $$are unit vectors, then$${\left| {\overrightarrow a - \overrightarrow b }...
IIT-JEE 2000 Screening
If the vectors $$\overrightarrow a ,\overrightarrow b$$ and $$\overrightarrow c$$ form the sides $$BC,$$ $$CA$$ and $$AB$$ respectively of a triangl...
IIT-JEE 2000 Screening
Let the vectors $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ and $$\overrightarrow d$$ be such that $$\left( {\overrightarrow a \ti... IIT-JEE 2000 Screening If$$\overrightarrow a \,,\,\overrightarrow b $$and$$\overrightarrow c $$are unit coplanar vectors, then the scalar triple product$$\left[ {2\over...
IIT-JEE 1999
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...
IIT-JEE 1999
Let $$a=2i+j+k, b=i+2j-k$$ and a unit vector $$c$$ be coplanar. If $$c$$ is perpendicular to $$a,$$ then $$c =$$
IIT-JEE 1998
If $$a = i + j + k,\overrightarrow b = 4i + 3j + 4k$$ and $$c = i + \alpha j + \beta k$$ are linearly dependent vectors and $$\left| c \right| = \sq... IIT-JEE 1998 For three vectors$$u,v,w$$which of the following expression is not equal to any of the remaining three? IIT-JEE 1995 Screening Let$$\overrightarrow a = \widehat i - \widehat j,\overrightarrow b = \widehat j - \widehat k,\overrightarrow c = \widehat k - \widehat i.$$If$$...
IIT-JEE 1995 Screening
If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ are non coplanar unit vectors such that $$\overrightarrow a \times \left( {\overright... IIT-JEE 1995 Screening Let$$\overrightarrow u ,\overrightarrow v $$and$$\overrightarrow w $$be vectors such that$$\overrightarrow u + \overrightarrow v + \overrightar...
IIT-JEE 1995 Screening
If $$\overrightarrow a ,$$ $$\overrightarrow b$$ and $$\overrightarrow c$$ are three non coplanar vectors, then $$\left( {\overrightarrow a + \ov... IIT-JEE 1993 Let$$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...
IIT-JEE 1988
Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,$$ be three non-coplanar vectors and $$\overrightarrow p ,\overrightarrow q ,\overright... IIT-JEE 1987 The number of vectors of unit length perpendicular to vectors$$\overrightarrow a = \left( {1,1,0} \right)$$and$$\overrightarrow b = \left( {0,1,1...
IIT-JEE 1986
Let $$\overrightarrow a = {a_1}i + {a_2}j + {a_3}k,\,\,\,\overrightarrow b = {b_1}i + {b_2}j + {b_3}k$$ and $$\overrightarrow c = {c_1}i + {c_2}j +... IIT-JEE 1982 For non-zero vectors$${\overrightarrow a ,\,\overrightarrow b ,\overrightarrow c },\left| {\left( {\overrightarrow a \times \overrightarrow b }...
IIT-JEE 1981
The scalar $$\overrightarrow A .\left( {\overrightarrow B + \overrightarrow C } \right) \times \left( {\overrightarrow A + \overrightarrow B + \ove... ## MCQ (More than One Correct Answer) JEE Advanced 2022 Paper 2 Online Let \hat{\imath}, \hat{\jmath} and \hat{k} be the unit vectors along the three positive coordinate axes. Let$$ \begin{aligned} & \vec{a}=3 \hat{...
JEE Advanced 2021 Paper 2 Online
Let O be the origin and $$\overrightarrow {OA} = 2\widehat i + 2\widehat j + \widehat k$$ and $$\overrightarrow {OB} = \widehat i - 2\widehat j + 2\... JEE Advanced 2020 Paper 2 Offline Let a and b be positive real numbers. Suppose$$PQ = a\widehat i + b\widehat j$$and$$PS = a\widehat i - b\widehat j$$are adjacent sides of a parall... JEE Advanced 2016 Paper 2 Offline Let$$\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...
JEE Advanced 2015 Paper 1 Offline
Let $$\Delta PQR$$ be a triangle. Let $$\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP}$$ and $$\overrightarrow c = \overrightarrow {PQ... JEE Advanced 2014 Paper 1 Offline Let$$\overrightarrow x ,\overrightarrow y $$and$$\overrightarrow z $$be three vectors each of magnitude$$\sqrt 2 $$and the angle between each pa... IIT-JEE 2011 Paper 1 Offline The vector (s) which is/are coplanar with vectors$${\widehat i + \widehat j + 2\widehat k}$$and$${\widehat i + 2\widehat j + \widehat k,}$$and per... IIT-JEE 1999 Let$$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|$$is IIT-JEE 1998 Which of the following expressions are meaningful? IIT-JEE 1994 The vector$$\,{1 \over 3}\left( {2\widehat i - 2\widehat j + \widehat k} \right)$$is IIT-JEE 1993 Let$$\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...

## Subjective

IIT-JEE 2006
Match the folowing : (A)$$\,\,\,$$Two rays $$x + y = \left| a \right|$$ and $$ax - y=1$$ intersects each other in the $$\,\,\,\,\,\,\,\,\,\,$$first q...
IIT-JEE 2005
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...
IIT-JEE 2004
If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ and $$\overrightarrow d$$ are distinct vectors such that $$\,\overrightarrow a \tim... IIT-JEE 2003 If$$\overrightarrow u ,\overrightarrow v ,\overrightarrow w ,$$are three non-coplanar unit vectors and$$\alpha ,\beta ,\gamma $$are the angles bet... IIT-JEE 2002 Let$$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,...
IIT-JEE 2001
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...
IIT-JEE 2001
Find $$3-$$dimensional vectors $${\overrightarrow v _1},{\overrightarrow v _2},{\overrightarrow v _3}$$ satisfying $$\,{\overrightarrow v _1}.{\over... IIT-JEE 2001 Let$$\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... IIT-JEE 1999 Let $$u$$ and $$v$$ be units vectors. If $$w$$ is a vector such that $$w + \left( {w \times u} \right) = v,$$ then prove that $$\left| {\left( {u \tim... IIT-JEE 1998 Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points ... IIT-JEE 1998 For any two vectors$$u$$and$$v,$$prove that (a)$${\left( {u\,.\,v} \right)^2} + {\left| {u \times v} \right|^2} = {\left| u \right|^2}{\left| v ... IIT-JEE 1997 If $$A,B$$ and $$C$$ are vectors such that $$\left| B \right| = \left| C \right|.$$ Prove that $$\left[ {\left( {A + B} \right) \times \left( {A + C}... IIT-JEE 1994 If the vectors$$\overrightarrow b ,\overrightarrow c ,\overrightarrow d ,$$are not coplanar, then prove that the vector$$\left( {\overrightarrow a... IIT-JEE 1993 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... IIT-JEE 1991 Determine the value of $$'c'$$ so that for all real $$x,$$ the vector $$cx\widehat i - 6\widehat j - 3\widehat k$$ and $$x\widehat i + 2\widehat j + ... IIT-JEE 1990 Let$$\overrightarrow A = 2\overrightarrow i + \overrightarrow k ,\,\overrightarrow B = \overrightarrow i + \overrightarrow j + \overrightarrow k... IIT-JEE 1989 If vectors $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C$$ are coplanar, show that $$\left| {\matrix{ {} & {\overrightarrow {a.}... IIT-JEE 1989 In a triangle$$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,$$det... IIT-JEE 1988 Let$$OACB$$be a parallelogram with$$O$$at the origin and$$OC$$a diagonal. Let$$D$$be the midpoint of$$OA.$$Using vector methods prove ... IIT-JEE 1987 If$$A, B, C, D$$are any four points in space, prove that -$$\left| {\overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \ti... IIT-JEE 1986 The position vectors of the points $$A, B, C$$ and $$D$$ are $$3\widehat i - 2\widehat j - \widehat k,\,2\widehat i + 3\widehat j - 4\widehat k,\, - \... IIT-JEE 1982$${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_... IIT-JEE 1982 Find all values of $$\lambda$$ such that $$x, y, z,$$$$\, \ne$$$$(0,0,0)$$ and $$\left( {\overrightarrow i + \overrightarrow j + 3\overrightarrow... ## Fill in the Blanks IIT-JEE 1997 Let$$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,$$an... IIT-JEE 1996 If$$\overrightarrow b \,$$and$$\overrightarrow c \,$$are two non-collinear unit vectors and$$\overrightarrow a \,$$is any vector, then$$\left( ... IIT-JEE 1996 A nonzero vector $$\overrightarrow a$$ is parallel to the line of intersection of the plane determined by the vectors $$\widehat i,\widehat i + \wide... IIT-JEE 1992 A unit vector coplanar with$$\overrightarrow i + \overrightarrow j + 2\overrightarrow k $$and$$\overrightarrow i + 2\overrightarrow j + \overri... IIT-JEE 1991 Given that $$\overrightarrow a = \left( {1,1,1} \right),\,\,\overrightarrow c = \left( {0,1, - 1} \right),\,\overrightarrow a .\overrightarrow b = ... IIT-JEE 1988 The components of a vector$$\overrightarrow a $$along and perpendicular to a non-zero vector$$\overrightarrow b $$are ......and .....respectively. IIT-JEE 1987 If the vectors$$a\widehat i + \widehat j + \widehat k,\,\,\widehat i + b\widehat j + \widehat k$$and$$\widehat i + \widehat j + c\widehat k\... IIT-JEE 1987 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... IIT-JEE 1985 If $$\overrightarrow A \overrightarrow {\,B} \overrightarrow {\,C}$$ are three non-coplannar vectors, then - $${{\overrightarrow A .\overrightarrow ... IIT-JEE 1985 If$$\overrightarrow A = \left( {1,1,1} \right),\,\,\overrightarrow C = \left( {0,1, - 1} \right)$$are given vectors, then a vector$$B$$satifying... IIT-JEE 1984$$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 - \...
IIT-JEE 1981
Let $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C$$ be vectors of length $$3, 4, 5$$ respectively. Let $$\overrightarrow A$$ be perpendi...

## True or False

IIT-JEE 1989
For any three vectors $${\overrightarrow a ,\,\overrightarrow b ,}$$ and $${\overrightarrow c ,}$$ $$\left( {\overrightarrow a - \overrightarrow b }... IIT-JEE 1984 The points with position vectors$$a+b,a-b,$$and$$a+kb$$are collinear for all real values of$$k.$$IIT-JEE 1983 If$$X.A=0, X.B=0, X.C=0$$for some non-zero vector$$X,$$then$$\left[ {A\,B\,C} \right] = 0$$IIT-JEE 1981 Let$$\overrightarrow A ,\overrightarrow B $$and$${\overrightarrow C }$$be unit vectors suppose that$$\overrightarrow A .\overrightarrow B = \ove...
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