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JEE Advanced

Vector Algebra and 3D Geometry

Mathematics

Previous Years Questions

MCQ (More than One Correct Answer)

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Let O be the origin and $$\overrightarrow {OA} = 2\widehat i + 2\widehat j + \widehat k$$ and $$\overrightarrow {OB} =...
JEE Advanced 2021 Paper 2 Online
Let a and b be positive real numbers. Suppose $$PQ = a\widehat i + b\widehat j$$ and $$PS = a\widehat i - b\widehat j$$ ...
JEE Advanced 2020 Paper 2 Offline
Let $$\alpha $$2 + $$\beta $$2 + $$\gamma $$2 $$ \ne $$ 0 and $$\alpha $$ + $$\gamma $$ = 1. Suppose the point (3, 2, $$...
JEE Advanced 2020 Paper 2 Offline
Let L1 and L2 be the following straight lines.$${L_1}:{{x - 1} \over 1} = {y \over { - 1}} = {{z - 1} \over 3}$$ and $${...
JEE Advanced 2020 Paper 1 Offline
Three lines $${L_1}:r = \lambda \widehat i$$, $$\lambda $$ $$ \in $$ R,$${L_2}:r = \widehat k + \mu \widehat j$$, $$\mu ...
JEE Advanced 2019 Paper 2 Offline
Let L1 and L2 denote the lines$$r = \widehat i + \lambda ( - \widehat i + 2\widehat j + 2\widehat k)$$, $$\lambda $$$$ \...
JEE Advanced 2019 Paper 1 Offline
Let P1 : 2x + y $$-$$ z = 3 and P2 : x + 2y + z = 2 be two planes. Then, which of the following statement(s) is(are) TR...
JEE Advanced 2018 Paper 1 Offline
Let $$\widehat u = {u_1} \widehat i + {u_2}\widehat j + {u_3}\widehat k$$ be a unit vector in $${{R^3}}$$ and $$\widehat...
JEE Advanced 2016 Paper 2 Offline
Consider a pyramid $$OPQRS$$ located in the first octant $$\left( {x \ge 0,y \ge 0,z \ge 0} \right)$$ with $$O$$ as orig...
JEE Advanced 2016 Paper 1 Offline
Let $$\Delta PQR$$ be a triangle. Let $$\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP} $$ and $$\overright...
JEE Advanced 2015 Paper 1 Offline
In $${R^3},$$ let $$L$$ be a straight lines passing through the origin. Suppose that all the points on $$L$$ are at a c...
JEE Advanced 2015 Paper 1 Offline
In $${R^3},$$ consider the planes $$\,{P_1}:y = 0$$ and $${P_2}:x + z = 1.$$ Let $${P_3}$$ be the plane, different from ...
JEE Advanced 2015 Paper 1 Offline
Let $$\overrightarrow x ,\overrightarrow y $$ and $$\overrightarrow z $$ be three vectors each of magnitude $$\sqrt 2 $$...
JEE Advanced 2014 Paper 1 Offline
Two lines $${L_1}:x = 5,{y \over {3 - \alpha }} = {z \over { - 2}}$$ and $${L_2}:x = \alpha ,{y \over { - 1}} = {z \over...
JEE Advanced 2013 Paper 2 Offline
A line $$l$$ passing through the origin is perpendicular to the lines $$$\,{l_1}:\left( {3 + t} \right)\widehat i + \le...
JEE Advanced 2013 Paper 1 Offline
If the straight lines $$\,{{x - 1} \over 2} = {{y + 1} \over k} = {z \over 2}$$ and $${{x + 1} \over 5} = {{y + 1} \over...
IIT-JEE 2012 Paper 2 Offline
The vector (s) which is/are coplanar with vectors $${\widehat i + \widehat j + 2\widehat k}$$ and $${\widehat i + 2\wide...
IIT-JEE 2011 Paper 1 Offline
Let $${\overrightarrow A }$$ be vector parallel to line of intersection of planes $${P_1}$$ and $${P_2}.$$ Planes $${P_...
IIT-JEE 2006
Let $$a$$ and $$b$$ two non-collinear unit vectors. If $$u = a - \left( {a\,.\,b} \right)\,b$$ and $$v = a \times b,$$ t...
IIT-JEE 1999
Which of the following expressions are meaningful?
IIT-JEE 1998
The vector $$\,{1 \over 3}\left( {2\widehat i - 2\widehat j + \widehat k} \right)$$ is
IIT-JEE 1994
Let $$\vec a = 2\hat i - \hat j + \hat k,\vec b = \hat i + 2\hat j - \hat k$$ and $$\overrightarrow c = \widehat i + \...
IIT-JEE 1993

Numerical

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Let $$\overrightarrow u $$, $$\overrightarrow v $$ and $$\overrightarrow w $$ be vectors in three-dimensional space, whe...
JEE Advanced 2021 Paper 1 Online
Let $$\alpha$$, $$\beta$$ and $$\gamma$$ be real numbers such that the system of linear equationsx + 2y + 3z = $$\alpha$...
JEE Advanced 2021 Paper 1 Online
Let $$\alpha$$, $$\beta$$ and $$\gamma$$ be real numbers such that the system of linear equationsx + 2y + 3z = $$\alpha$...
JEE Advanced 2021 Paper 1 Online
Let $$\overrightarrow a = 2\widehat i + \widehat j - \widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j + \...
JEE Advanced 2019 Paper 2 Offline
Three lines are given by $$r = \lambda \widehat i,\,\lambda \in R$$, $$r = \mu (\widehat i + \widehat j),\,\mu \in R$$...
JEE Advanced 2019 Paper 1 Offline
Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicula...
JEE Advanced 2018 Paper 2 Offline
Let a and b be two unit vectors such that a . b = 0. For some x, y$$ \in $$R, let $$\overrightarrow c = x\overrightarro...
JEE Advanced 2018 Paper 1 Offline
Suppose that $$\overrightarrow p ,\overrightarrow q $$ and $$\overrightarrow r $$ are three non-coplanar vectors in $${R...
JEE Advanced 2015 Paper 2 Offline
Let $$\overrightarrow a \,\,,\,\,\overrightarrow b $$ and $$\overrightarrow c $$ be three non-coplanar unit vectors such...
JEE Advanced 2014 Paper 1 Offline
Consider the set of eight vectors $$V = \left\{ {a\widehat i + b\widehat j + c\widehat k:a,b.c \in \left\{ { - 1,1} \ri...
JEE Advanced 2013 Paper 1 Offline
If $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ are unit vectors satisfying $${\left| {\overrig...
IIT-JEE 2012 Paper 1 Offline
Let $$\overrightarrow a = - \widehat i - \widehat k,\overrightarrow b = - \widehat i + \widehat j$$ and $$\overright...
IIT-JEE 2011 Paper 2 Offline
If the distance between the plane $$Ax-2y+z=d$$ and the plane containing the lines $${{x - 1} \over 2} = {{y - 2} \over ...
IIT-JEE 2010 Paper 1 Offline
If $$\overrightarrow a $$ and $$\overrightarrow b $$ are vectors in space given by $$\overrightarrow a = {{\widehat i -...
IIT-JEE 2010 Paper 1 Offline

MCQ (Single Correct Answer)

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|$$\overrightarrow{OX}$$ $$ \times $$ $$\overrightarrow{OY}$$| = ?
JEE Advanced 2017 Paper 2 Offline
If the triangle PQR varies, then the minimum value of cos(P + Q) + cos(Q + R) + cos(R + P) is
JEE Advanced 2017 Paper 2 Offline
The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes 2x + y $$-$$ 2z = 5 and 3x...
JEE Advanced 2017 Paper 2 Offline
Let O be the origin and let PQR be an arbitrary triangle. The point S is such that$$\overrightarrow{OP}$$ . $$\overright...
JEE Advanced 2017 Paper 2 Offline
Let $$P$$ be the image of the point $$(3,1,7)$$ with respect to the plane $$x-y+z=3.$$ Then the equation of the plane pa...
JEE Advanced 2016 Paper 2 Offline
Match the following : $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column $$I$$ (A)$$\,\,\,\,$$ In a triangle $$\Delta XYZ,$...
JEE Advanced 2015 Paper 1 Offline
Match the following : $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column $$I$$ (A)$$\,\,\,\,$$ In $${R^2},$$ If the magnitu...
JEE Advanced 2015 Paper 1 Offline
From a point $$P\left( {\lambda ,\lambda ,\lambda } \right),$$ perpendicular $$PQ$$ and $$PR$$ are drawn respectively on...
JEE Advanced 2014 Paper 1 Offline
Consider the lines $${L_1}:{{x - 1} \over 2} = {y \over { - 1}} = {{z + 3} \over 1},{L_2} : {{x - 4} \over 1} = {{y + 3}...
JEE Advanced 2013 Paper 2 Offline
match List $$I$$ with List $$II$$ and select the correct answer using the code given below the lists: $$\,\,\,\,$$ $$\...
JEE Advanced 2013 Paper 2 Offline
If $$\overrightarrow a $$ and $$\overrightarrow b $$ are vectors such that $$\left| {\overrightarrow a + \overrightarr...
IIT-JEE 2012 Paper 2 Offline
The point $$P$$ is the intersection of the straight line joining the points $$Q(2, 3, 5)$$ and $$R(1, -1, 4)$$ with the ...
IIT-JEE 2012 Paper 1 Offline
The equation of a plane passing through the line of intersection of the planes $$x+2y+3z=2$$ and $$x-y+z=3$$ and at a di...
IIT-JEE 2012 Paper 2 Offline
Match the statements given in Column -$$I$$ with the values given in Column-$$II.$$ $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,...
IIT-JEE 2011 Paper 2 Offline
Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \wideha...
IIT-JEE 2011 Paper 1 Offline
Match the statement in Column-$$I$$ with the values in Column-$$II$$ $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I...
IIT-JEE 2010 Paper 2 Offline
Equation of the plane containing the straight line $${x \over 2} = {y \over 3} = {z \over 4}$$ and perpendicular to the ...
IIT-JEE 2010 Paper 1 Offline
If the distance of the point $$P(1, -2, 1)$$ from the plane $$x+2y-2z$$$$\, = \alpha ,$$ where $$\alpha > 0,$$ is $...
IIT-JEE 2010 Paper 2 Offline
Two adjacent sides of a parallelogram $$ABCD$$ are given by $$\overrightarrow {AB} = 2\widehat i + 10\widehat j + 11\w...
IIT-JEE 2010 Paper 2 Offline
Let $$P,Q,R$$ and $$S$$ be the points on the plane with position vectors $${ - 2\widehat i - \widehat j,4\widehat i,3\wi...
IIT-JEE 2010 Paper 1 Offline
A line with positive direction cosines passes through the point $$P(2, -1, 2)$$ and makes equal angles with the coordina...
IIT-JEE 2009
Let $$P(3,2,6)$$ be a point in space and $$Q$$ be a point on the line $$$\widehat r = \left( {\widehat i - \widehat j +...
IIT-JEE 2009
If $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are unit vectors such tha...
IIT-JEE 2009
Consider three planes $$${P_1}:x - y + z = 1$$$ $$${P_2}:x + y - z = 1$$$ $$${P_3}:x - 3y + 3z = 2$$$ Let $${L_1},$$ $$...
IIT-JEE 2008
Consider the lines $${L_1}:{{x + 1} \over 3} = {{y + 2} \over 1} = {{z + 1} \over 2}\,\,\,\,{L_2}:{{x - 2} \over 1} = ...
IIT-JEE 2008
Consider the lines $${L_1}:{{x + 1} \over 3} = {{y + 2} \over 1} = {{z + 1} \over 2}\,\,\,\,{L_2}:{{x - 2} \over 1} = {...
IIT-JEE 2008
Consider the lines $${L_1}:{{x + 1} \over 3} = {{y + 2} \over 1} = {{z + 1} \over 2}\,\,\,\,{L_2}:{{x - 2} \over 1} = {...
IIT-JEE 2008
Let two non-collinear unit vectors $$\widehat a$$ and $$\widehat b$$ form an acute angle. A point $$P$$ moves so that at...
IIT-JEE 2008
The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $$\overrightarrow a \,,\,...
IIT-JEE 2008
Let the vectors $$\overrightarrow {PQ} ,\,\,\overrightarrow {QR} ,\,\,\overrightarrow {RS} ,\,\,\overrightarrow {ST} ,\,...
IIT-JEE 2007
Consider the planes $$3x-6y-2z=15$$ and $$2x+y-2z=5.$$ STATEMENT-1: The parametric equations of the line of intersectio...
IIT-JEE 2007
Let $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ be unit vectors such that $${\overrightarrow a + ...
IIT-JEE 2007
The minimum of distinct real values of $$\lambda ,$$ for which the vectors $$ - {\lambda ^2}\widehat i + \widehat j + \w...
IIT-JEE 2007
A plane which is perpendicular to two planes $$2x - 2y + z = 0$$ and $$x - y + 2z = 4,$$ passes through $$(1, -2, 1).$$ ...
IIT-JEE 2006
Let $$\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \wideh...
IIT-JEE 2006
If $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ are three non-zero, non-coplanar vectors and $$\ov...
IIT-JEE 2005 Screening
A variable plane at a distance of the one unit from the origin cuts the coordinates axes at $$A,$$ $$B$$ and $$C.$$ If t...
IIT-JEE 2005 Screening
The unit vector which is orthogonal to the vector $$3\overrightarrow i + 2\overrightarrow j + 6\overrightarrow k $$ an...
IIT-JEE 2004 Screening
If the lines $${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4}$$ and $$\,{{x - 3} \over 1} = {{y - k} \over 2}...
IIT-JEE 2004 Screening
If $$\overrightarrow a = \left( {\widehat i + \widehat j + \widehat k} \right),\overrightarrow a .\overrightarrow b = ...
IIT-JEE 2004 Screening
The value of $$'a'$$ so that the volume of parallelopiped formed by $$\widehat i + a\widehat j + \widehat k,\widehat j +...
IIT-JEE 2003 Screening
The value of $$k$$ such that $${{x - 4} \over 1} = {{y - 2} \over 1} = {{z - k} \over 2}$$ lies in the plane $$2x -4y +z...
IIT-JEE 2003 Screening
Let $$\overrightarrow V = 2\overrightarrow i + \overrightarrow j - \overrightarrow k $$ and $$\overrightarrow W = \o...
IIT-JEE 2002 Screening
If $${\overrightarrow a }$$ and $${\overrightarrow b }$$ are two unit vectors such that $${\overrightarrow a + 2\overri...
IIT-JEE 2002 Screening
If $$\overrightarrow a \,,\,\overrightarrow b $$ and $$\overrightarrow c $$ are unit vectors, then $${\left| {\overright...
IIT-JEE 2001 Screening
Let $$\overrightarrow a = \overrightarrow i - \overrightarrow k ,\overrightarrow b = x\overrightarrow i + \overright...
IIT-JEE 2001 Screening
If $$\overrightarrow a \,,\,\overrightarrow b $$ and $$\overrightarrow c $$ are unit coplanar vectors, then the scalar t...
IIT-JEE 2000 Screening
Let the vectors $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ be such that $$...
IIT-JEE 2000 Screening
If the vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ form the sides $$BC,$$ $$CA$$ and $$...
IIT-JEE 2000 Screening
Let $$a=2i+j+k, b=i+2j-k$$ and a unit vector $$c$$ be coplanar. If $$c$$ is perpendicular to $$a,$$ then $$c =$$
IIT-JEE 1999
Let $$a=2i+j-2k$$ and $$b=i+j.$$ If $$c$$ is a vector such that $$a.$$ $$c = \left| c \right|,\left| {c - a} \right| = ...
IIT-JEE 1999
For three vectors $$u,v,w$$ which of the following expression is not equal to any of the remaining three?
IIT-JEE 1998
If $$a = i + j + k,\overrightarrow b = 4i + 3j + 4k$$ and $$c = i + \alpha j + \beta k$$ are linearly dependent vector...
IIT-JEE 1998
If $$\overrightarrow a ,$$ $$\overrightarrow b $$ and $$\overrightarrow c $$ are three non coplanar vectors, then $$\l...
IIT-JEE 1995 Screening
Let $$\overrightarrow u ,\overrightarrow v $$ and $$\overrightarrow w $$ be vectors such that $$\overrightarrow u + \ov...
IIT-JEE 1995 Screening
If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ are non coplanar unit vectors such that $$\overrightarro...
IIT-JEE 1995 Screening
Let $$\overrightarrow a = \widehat i - \widehat j,\overrightarrow b = \widehat j - \widehat k,\overrightarrow c = \wi...
IIT-JEE 1995 Screening
Let $$\overrightarrow p $$ and $$\overrightarrow q $$ be the position vectors of $$P$$ and $$Q$$ respectively, with resp...
IIT-JEE 1994
Let $$\alpha ,\beta ,\gamma $$ be distinct real numbers. The points with position vectors $$\alpha \widehat i + \beta \...
IIT-JEE 1994
Let $$a, b, c$$ be distinct non-negative numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\widehat i + ...
IIT-JEE 1993
Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,$$ be three non-coplanar vectors and $$\overrightarrow p ...
IIT-JEE 1988
The number of vectors of unit length perpendicular to vectors $$\overrightarrow a = \left( {1,1,0} \right)$$ and $$\ove...
IIT-JEE 1987
Let $$\overrightarrow a = {a_1}i + {a_2}j + {a_3}k,\,\,\,\overrightarrow b = {b_1}i + {b_2}j + {b_3}k$$ and $$\overrig...
IIT-JEE 1986
The points with position vectors $$60i+3j,$$ $$40i-8j,$$ $$ai-52j$$ are collinear if
IIT-JEE 1983
The volume of the parallelopiped whose sides are given by $$\overrightarrow {OA} = 2i - 2j,\,\overrightarrow {OB} = i...
IIT-JEE 1983
For non-zero vectors $${\overrightarrow a ,\,\overrightarrow b ,\overrightarrow c },$$ $$\left| {\left( {\overrightarrow...
IIT-JEE 1982
The scalar $$\overrightarrow A .\left( {\overrightarrow B + \overrightarrow C } \right) \times \left( {\overrightarrow ...
IIT-JEE 1981

Subjective

More
Match the statements/expressions given in Column-$$I$$ with the values given in Column-$$II.$$ $$\,\,\,\,\,$$ $$\,\,\,\,...
IIT-JEE 2009
Match the statements / expressions given in Column-$$I$$ with the values given in Column-$$II.$$ $$\,\,\,\,$$ $$\,\,\,\...
IIT-JEE 2009
Consider the following linear equations $$ax+by+cz=0;$$ $$\,\,\,$$ $$bx+cy+az=0;$$ $$\,\,\,$$ $$cx+ay+bz=0$$ Match the ...
IIT-JEE 2007
Match the folowing : (A)$$\,\,\,$$Two rays $$x + y = \left| a \right|$$ and $$ax - y=1$$ intersects each other in the $...
IIT-JEE 2006
If the incident ray on a surface is along the unit vector $$\widehat v\,\,,$$ the reflected ray is along the unit vector...
IIT-JEE 2005
Find the equation of the plane containing the line $$2x-y+z-3=0,3x+y+z=5$$ and at a distance of $${1 \over {\sqrt 6 }}$$...
IIT-JEE 2005
$${P_1}$$ and $${P_2}$$ are planes passing through origin. $${L_1}$$ and $${L_2}$$ are two line on $${P_1}$$ and $${P_2}...
IIT-JEE 2004
If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are distinct vectors such tha...
IIT-JEE 2004
A parallelopiped $$'S'$$ has base points $$A, B, C$$ and $$D$$ and upper face points $$A',$$ $$B',$$ $$C'$$ and $$D'.$$ ...
IIT-JEE 2004
Find the equation of plane passing through $$(1, 1, 1)$$ & parallel to the lines $${L_1},{L_2}$$ having direction ra...
IIT-JEE 2004
(i) Find the equation of the plane passing through the points $$(2, 1, 0), (5, 0, 1)$$ and $$(4, 1, 1).$$ (ii) If $$P$$...
IIT-JEE 2003
If $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w ,$$ are three non-coplanar unit vectors and $$\alpha ,\beta...
IIT-JEE 2003
Let $$V$$ be the volume of the parallelopiped formed by the vectors $$\overrightarrow a = {a_1}\widehat i + {a_2}\wideh...
IIT-JEE 2002
Let $$\overrightarrow A \left( t \right) = {f_1}\left( t \right)\widehat i + {f_2}\left( t \right)\widehat j$$ and $$$\...
IIT-JEE 2001
Find $$3-$$dimensional vectors $${\overrightarrow v _1},{\overrightarrow v _2},{\overrightarrow v _3}$$ satisfying $$\...
IIT-JEE 2001
Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position...
IIT-JEE 2001
Let $$u$$ and $$v$$ be units vectors. If $$w$$ is a vector such that $$w + \left( {w \times u} \right) = v,$$ then prove...
IIT-JEE 1999
For any two vectors $$u$$ and $$v,$$ prove that (a) $${\left( {u\,.\,v} \right)^2} + {\left| {u \times v} \right|^2} = ...
IIT-JEE 1998
Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line p...
IIT-JEE 1998
If $$A,B$$ and $$C$$ are vectors such that $$\left| B \right| = \left| C \right|.$$ Prove that $$\left[ {\left( {A + B}...
IIT-JEE 1997
The position vectors of the vertices $$A, B$$ and $$C$$ of a tetrahedron $$ABCD$$ are $$\widehat i + \widehat j + \wideh...
IIT-JEE 1996
If the vectors $$\overrightarrow b ,\overrightarrow c ,\overrightarrow d ,$$ are not coplanar, then prove that the vecto...
IIT-JEE 1994
In a triangle $$ABC, D$$ and $$E$$ are points on $$BC$$ and $$AC$$ respectively, such that $$BD=2DC$$ and $$AE=3EC.$$ Le...
IIT-JEE 1993
Determine the value of $$'c'$$ so that for all real $$x,$$ the vector $$cx\widehat i - 6\widehat j - 3\widehat k$$ and ...
IIT-JEE 1991
Let $$\overrightarrow A = 2\overrightarrow i + \overrightarrow k ,\,\overrightarrow B = \overrightarrow i + \overrig...
IIT-JEE 1990
In a triangle $$OAB,E$$ is the midpoint of $$BO$$ and $$D$$ is a point on $$AB$$ such that $$AD:DB=2:1.$$ If $$OD$$ and ...
IIT-JEE 1989
If vectors $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$ are coplanar, show that $$$\left| {\matrix{ ...
IIT-JEE 1989
Let $$OA$$ $$CB$$ be a parallelogram with $$O$$ at the origin and $$OC$$ a diagonal. Let $$D$$ be the midpoint of $$OA....
IIT-JEE 1988
If $$A, B, C, D$$ are any four points in space, prove that - $$\left| {\overrightarrow {AB} \times \overrightarrow {CD...
IIT-JEE 1987
The position vectors of the points $$A, B, C$$ and $$D$$ are $$3\widehat i - 2\widehat j - \widehat k,\,2\widehat i + 3\...
IIT-JEE 1986
A vector $$\overrightarrow A $$ has components $${A_1},{A_2},{A_3}$$ in a right -handed rectangular Cartesian coordinate...
IIT-JEE 1983
Find all values of $$\lambda $$ such that $$x, y, z,$$$$\, \ne $$$$(0,0,0)$$ and $$\left( {\overrightarrow i + \overri...
IIT-JEE 1982
$${A_1},{A_2},.................{A_n}$$ are the vertices of a regular plane polygon with $$n$$ sides and $$O$$ is its cen...
IIT-JEE 1982
From a point $$O$$ inside a triangle $$ABC,$$ perpendiculars $$OD$$, $$OE, OF$$ are drawn to the sides $$BC, CA, AB$$ re...
IIT-JEE 1978

Fill in the Blanks

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Let $$OA=a,$$ $$OB=10a+2b$$ and $$OC=b$$ where $$O,A$$ and $$C$$ are non-collinear points. Let $$p$$ denote the area of ...
IIT-JEE 1997
If $$\overrightarrow b \,$$ and $$\overrightarrow c \,$$ are two non-collinear unit vectors and $$\overrightarrow a \,$$...
IIT-JEE 1996
A nonzero vector $$\overrightarrow a $$ is parallel to the line of intersection of the plane determined by the vectors $...
IIT-JEE 1996
A unit vector perpendicular to the plane determined by the points $$P\left( {1, - 1,2} \right)Q\left( {2,0, - 1} \right)...
IIT-JEE 1994
A unit vector coplanar with $$\overrightarrow i + \overrightarrow j + 2\overrightarrow k $$ and $$\overrightarrow i +...
IIT-JEE 1992
Given that $$\overrightarrow a = \left( {1,1,1} \right),\,\,\overrightarrow c = \left( {0,1, - 1} \right),\,\overright...
IIT-JEE 1991
The components of a vector $$\overrightarrow a $$ along and perpendicular to a non-zero vector $$\overrightarrow b $$ ar...
IIT-JEE 1988
Let $$b = 4\widehat i + 3\widehat j$$ and $$\overrightarrow c $$ be two vectors perpendicular to each other in the $$xy$...
IIT-JEE 1987
If the vectors $$a\widehat i + \widehat j + \widehat k,\,\,\widehat i + b\widehat j + \widehat k$$ and $$\widehat i + \...
IIT-JEE 1987
If $$\overrightarrow A = \left( {1,1,1} \right),\,\,\overrightarrow C = \left( {0,1, - 1} \right)$$ are given vectors,...
IIT-JEE 1985
If $$\overrightarrow A \overrightarrow {\,B} \overrightarrow {\,C} $$ are three non-coplannar vectors, then - $${{\over...
IIT-JEE 1985
If $$\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {...
IIT-JEE 1985
$$A, B, C$$ and $$D,$$ are four points in a plane with position vectors $$a, b, c$$ and $$d$$ respectively such that $$$...
IIT-JEE 1984
The area of the triangle whose vertices are $$A(1, -1, 2), B(2, 1, -1), C(3, -1, 2)$$ is ..........
IIT-JEE 1983
The unit vector perpendicular to the plane determined by $$P\left( {1, - 1,2} \right),\,Q\left( {2,0, - 1} \right)$$ and...
IIT-JEE 1983
Let $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$ be vectors of length $$3, 4, 5$$ respectively. Let $$\o...
IIT-JEE 1981

True or False

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

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