2003
JEE Advanced
Numerical
IIT-JEE 2003
If $\overrightarrow u ,\overrightarrow v ,\overrightarrow w ,$ are three non-coplanar unit vectors and $\alpha ,\beta ,\gamma $ are the angles between $\overrightarrow u $ and $\overrightarrow v $ and $\overrightarrow w ,$ $\overrightarrow w $ and $\overrightarrow u $ respectively and $\overrightarrow x ,\overrightarrow y ,\overrightarrow z ,$ are unit vectors along the bisectors of the angles $\alpha ,\,\,\beta ,\,\,\gamma $ respectively. Prove that $\,\left[ {\overrightarrow x \times \overrightarrow y \,\,\overrightarrow y \times \overrightarrow z \,\,\overrightarrow z \times \overrightarrow x } \right] = {1 \over {16}}{\left[ {\overrightarrow u \,\,\overrightarrow v \,\,\overrightarrow w } \right]^2}\,{\sec ^2}{\alpha \over 2}{\sec ^2}{\beta \over 2}{\sec ^2}{\gamma \over 2}.$
Correct Answer: Solve it.
2002
JEE Advanced
Numerical
IIT-JEE 2002
Let $V$ be the volume of the parallelopiped formed by the vectors $\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k,$ $\,\,\,\,\overrightarrow b = {b_1}\widehat i + {b_2}\widehat j + {b_3}\widehat k,$ $\,\,\,\,\,\overrightarrow c = {c_1}\widehat i + {c_2}\widehat j + {c_3}\widehat k.$ where $r=1, 2, 3,$ are non-negative real numbers and $\sum\limits_{r = 1}^3 {\left( {{a_r} + {b_r} + {c_r}} \right) = 3L,} $ show that $V \le {L^3}\,\,.$
Correct Answer: Solve it.
2001
JEE Advanced
Numerical
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 concurrency in terms of the position vectors of the vertices.
Correct Answer: Solve it.
2001
JEE Advanced
Numerical
IIT-JEE 2001
Find $3-$dimensional vectors ${\overrightarrow v _1},{\overrightarrow v _2},{\overrightarrow v _3}$ satisfying
$\,{\overrightarrow v _1}.{\overrightarrow v _1} = 4,\,{\overrightarrow v _1}.{\overrightarrow v _2} = - 2,\,{\overrightarrow v _1}.{\overrightarrow v _3} = 6,\,\,{\overrightarrow v _2}.{\overrightarrow v _2}$
$ = 2,\,{\overrightarrow v _2}.{\overrightarrow v _3} = - 5,\,{\overrightarrow v _3}.{\overrightarrow v _3} = 29$
$\,{\overrightarrow v _1}.{\overrightarrow v _1} = 4,\,{\overrightarrow v _1}.{\overrightarrow v _2} = - 2,\,{\overrightarrow v _1}.{\overrightarrow v _3} = 6,\,\,{\overrightarrow v _2}.{\overrightarrow v _2}$
$ = 2,\,{\overrightarrow v _2}.{\overrightarrow v _3} = - 5,\,{\overrightarrow v _3}.{\overrightarrow v _3} = 29$
Correct Answer: $${\overrightarrow V _1} = 2\widehat i\,;\,\,\,{\overrightarrow V _2} = - \widehat i \pm \widehat j\,;\,\,\,{\overrightarrow V _3} = 3\widehat i \pm 2\widehat j \pm 4\widehat k$$
2001
JEE Advanced
Numerical
IIT-JEE 2001
Let $\overrightarrow A \left( t \right) = {f_1}\left( t \right)\widehat i + {f_2}\left( t \right)\widehat j$ and
$$\overrightarrow B \left( t \right) = {g_1}\left( t \right)\overrightarrow i + {g_2}\left( t \right)\widehat j,t \in \left[ {0,1} \right],$$
where ${f_1},{f_2},{g_1},{g_2}$ are continuous functions. If $\overrightarrow A \left( t \right)$ and $\overrightarrow B \left( t \right)$ are nonzero vectors for all $t$ and $\overrightarrow A \left( 0 \right) = 2\widehat i + 3\widehat j,$ $\,\overrightarrow A \left( 1 \right) = 6\widehat i + 2\widehat j,$ $\,\overrightarrow B \left( 0 \right) = 3\widehat i + 2\widehat j$ and $\,\overrightarrow B \left( 1 \right) = 2\widehat i + 6\widehat j.$ Then show that $\,\overrightarrow A \left( t \right)$ and $\,\overrightarrow B \left( t \right)$ are parallel for some $t.$
where ${f_1},{f_2},{g_1},{g_2}$ are continuous functions. If $\overrightarrow A \left( t \right)$ and $\overrightarrow B \left( t \right)$ are nonzero vectors for all $t$ and $\overrightarrow A \left( 0 \right) = 2\widehat i + 3\widehat j,$ $\,\overrightarrow A \left( 1 \right) = 6\widehat i + 2\widehat j,$ $\,\overrightarrow B \left( 0 \right) = 3\widehat i + 2\widehat j$ and $\,\overrightarrow B \left( 1 \right) = 2\widehat i + 6\widehat j.$ Then show that $\,\overrightarrow A \left( t \right)$ and $\,\overrightarrow B \left( t \right)$ are parallel for some $t.$
Correct Answer: Solve it.
1999
JEE Advanced
Numerical
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 $\left| {\left( {u \times v} \right) \cdot w} \right| \le 1/2$ and that the equality holds if and only if $u$ is perpendicular to $v .$
Correct Answer: Solve it.
1998
JEE Advanced
Numerical
IIT-JEE 1998
For any two vectors $u$ and $v,$ prove that
(a) ${\left( {u\,.\,v} \right)^2} + {\left| {u \times v} \right|^2} = {\left| u \right|^2}{\left| v \right|^2}$ and
(b) $\left( {1 + {{\left| u \right|}^2}} \right)\left( {1 + {{\left| v \right|}^2}} \right) = {\left( {1 - u.v} \right)^2} + {\left| {u + v + \left( {u \times v} \right)} \right|^2}.$
(a) ${\left( {u\,.\,v} \right)^2} + {\left| {u \times v} \right|^2} = {\left| u \right|^2}{\left| v \right|^2}$ and
(b) $\left( {1 + {{\left| u \right|}^2}} \right)\left( {1 + {{\left| v \right|}^2}} \right) = {\left( {1 - u.v} \right)^2} + {\left| {u + v + \left( {u \times v} \right)} \right|^2}.$
Correct Answer: Solve it.
1998
JEE Advanced
Numerical
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 of the parallel sides. (You may assume that the trapezium is not a parallelogram.)
Correct Answer: Solve it.
1997
JEE Advanced
Numerical
IIT-JEE 1997
If $A,B$ and $C$ are vectors such that $\left| B \right| = \left| C \right|.$ Prove that
$\left[ {\left( {A + B} \right) \times \left( {A + C} \right)} \right] \times \left( {B \times C} \right)\left( {B + C} \right) = 0\,\,.$
$\left[ {\left( {A + B} \right) \times \left( {A + C} \right)} \right] \times \left( {B \times C} \right)\left( {B + C} \right) = 0\,\,.$
Correct Answer: Solve it.
1994
JEE Advanced
Numerical
IIT-JEE 1994
If the vectors $\overrightarrow b ,\overrightarrow c ,\overrightarrow d ,$ are not coplanar, then prove that the vector
$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) + \left( {\overrightarrow a \times \overrightarrow c } \right) \times \left( {\overrightarrow d \times \overrightarrow b } \right) + \left( {\overrightarrow a \times \overrightarrow d } \right) \times \left( {\overrightarrow b \times \overrightarrow c } \right)$ is parallel to $\overrightarrow a .$
$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) + \left( {\overrightarrow a \times \overrightarrow c } \right) \times \left( {\overrightarrow d \times \overrightarrow b } \right) + \left( {\overrightarrow a \times \overrightarrow d } \right) \times \left( {\overrightarrow b \times \overrightarrow c } \right)$ is parallel to $\overrightarrow a .$
Correct Answer: Solve it.
1993
JEE Advanced
Numerical
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 intersection of $AD$ and $BE.$ Find $BP/PE$ using vector methods.
Correct Answer: $$8:3$$
1991
JEE Advanced
Numerical
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 $x\widehat i + 2\widehat j + 2cx\widehat k$ make an obtuse angle with each other.
$cx\widehat i - 6\widehat j - 3\widehat k$ and $x\widehat i + 2\widehat j + 2cx\widehat k$ make an obtuse angle with each other.
Correct Answer: $$ - {4 \over 3} < C < 0$$
1990
JEE Advanced
Numerical
IIT-JEE 1990
Let $\overrightarrow A = 2\overrightarrow i + \overrightarrow k ,\,\overrightarrow B = \overrightarrow i + \overrightarrow j + \overrightarrow k ,$ and $\overrightarrow C = 4\overrightarrow i - 3\overrightarrow j + 7\overrightarrow k .$ Determine a vector $\overrightarrow R .$ Satisfying $\overrightarrow R \times \overrightarrow B = \overrightarrow C \times \overrightarrow B $ and $\overrightarrow R \,.\,\overrightarrow A = 0$
Correct Answer: $$ - \widehat i - 8\widehat j + 2\widehat k$$
1989
JEE Advanced
Numerical
IIT-JEE 1989
If vectors $\overrightarrow A ,\overrightarrow B ,\overrightarrow C $ are coplanar, show that
$$\left| {\matrix{
{} & {\overrightarrow {a.} } & {} & {\overrightarrow {b.} } & {} & {\overrightarrow {c.} } \cr
{\overrightarrow {a.} } & {\overrightarrow {a.} } & {\overrightarrow {a.} } & {\overrightarrow {b.} } & {\overrightarrow {a.} } & {\overrightarrow {c.} } \cr
{\overrightarrow {b.} } & {\overrightarrow {a.} } & {\overrightarrow {b.} } & {\overrightarrow {b.} } & {\overrightarrow {b.} } & {\overrightarrow {c.} } \cr
} } \right| = \overrightarrow 0 $$
Correct Answer: Solve it.
1989
JEE Advanced
Numerical
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 $P,$ determine the ratio $OP:PD$ using vector methods.
Correct Answer: $$3:2$$
1988
JEE Advanced
Numerical
IIT-JEE 1988
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 that $BD$ and $CO$ intersect in the same ratio. Determine this ratio.
Correct Answer: $$2:1$$
1987
JEE Advanced
Numerical
IIT-JEE 1987
If $A, B, C, D$ are any four points in space, prove that -
$\left| {\overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \times \overrightarrow {AD} + \overrightarrow {CA} \times \overrightarrow {BD} } \right| = 4$ (area of triangle $ABC$)
$\left| {\overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \times \overrightarrow {AD} + \overrightarrow {CA} \times \overrightarrow {BD} } \right| = 4$ (area of triangle $ABC$)
Correct Answer: Solve it.
1986
JEE Advanced
Numerical
IIT-JEE 1986
The position vectors of the points $A, B, C$ and $D$ are $3\widehat i - 2\widehat j - \widehat k,\,2\widehat i + 3\widehat j - 4\widehat k,\, - \widehat i + \widehat j + 2\widehat k$ and $4\widehat i + 5\widehat j + \lambda \widehat k,$
respectively. If the points $A, B, C$ and $D$ lie on a plane, find the value of $\lambda .$
respectively. If the points $A, B, C$ and $D$ lie on a plane, find the value of $\lambda .$
Correct Answer: $${{146} \over {17}}$$
1982
JEE Advanced
Numerical
IIT-JEE 1982
Find all values of $\lambda $ such that $x, y, z,$$\, \ne $$(0,0,0)$ and
$\left( {\overrightarrow i + \overrightarrow j + 3\overrightarrow k } \right)x + \left( {3\overrightarrow i - 3\overrightarrow j + \overrightarrow k } \right)y + \left( { - 4\overrightarrow i + 5\overrightarrow j } \right)z$
$ = \lambda \left( {x\overrightarrow i \times \overrightarrow j \,\,y + \overrightarrow k \,z} \right)$ where $\overrightarrow i ,\,\,\overrightarrow j ,\,\,\overrightarrow k $ are unit vectors along the coordinate axes.
$\left( {\overrightarrow i + \overrightarrow j + 3\overrightarrow k } \right)x + \left( {3\overrightarrow i - 3\overrightarrow j + \overrightarrow k } \right)y + \left( { - 4\overrightarrow i + 5\overrightarrow j } \right)z$
$ = \lambda \left( {x\overrightarrow i \times \overrightarrow j \,\,y + \overrightarrow k \,z} \right)$ where $\overrightarrow i ,\,\,\overrightarrow j ,\,\,\overrightarrow k $ are unit vectors along the coordinate axes.
Correct Answer: $$\lambda = 0,\, - 1$$
1982
JEE Advanced
Numerical
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_{i = 1}^{n - 1} {\left( {\overrightarrow {O{A_i}} \times {{\overrightarrow {OA} }_{i + 1}}} \right) = \left( {1 - n} \right)\left( {{{\overrightarrow {OA} }_2} \times {{\overrightarrow {OA} }_1}} \right)} $
$\sum\limits_{i = 1}^{n - 1} {\left( {\overrightarrow {O{A_i}} \times {{\overrightarrow {OA} }_{i + 1}}} \right) = \left( {1 - n} \right)\left( {{{\overrightarrow {OA} }_2} \times {{\overrightarrow {OA} }_1}} \right)} $
Correct Answer: Solve it
1997
JEE Advanced
Numerical
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 $OABC,$ and let $q$ denote the area of the parallelogram with $OA$ and $OC$ as adjacent sides. If $p=kq,$ then $k=$.........
Correct Answer: $$6$$
1996
JEE Advanced
Numerical
IIT-JEE 1996
If $\overrightarrow b \,$ and $\overrightarrow c \,$ are two non-collinear unit vectors and $\overrightarrow a \,$ is any vector, then $\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow b + \left( {\overrightarrow a .\overrightarrow c } \right)\overrightarrow c + {{\overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)} \over {\left| {\overrightarrow b \times \overrightarrow c } \right|}}\left( {\overrightarrow b \times \overrightarrow c } \right) = $ ..............
Correct Answer: $${\overrightarrow a }$$
1996
JEE Advanced
Numerical
IIT-JEE 1996
A nonzero vector $\overrightarrow a $ is parallel to the line of intersection of the plane determined by the vectors $\widehat i,\widehat i + \widehat j$ and the plane determined by the vectors $\widehat i - \widehat j,\widehat i + \widehat k.$ The angle between $\overrightarrow a $ and the vector $\widehat i - 2\widehat j + 2\widehat k$ is ................
Correct Answer: $${\pi \over 4}$$ or $${3\pi \over 4}$$
1992
JEE Advanced
Numerical
IIT-JEE 1992
A unit vector coplanar with $\overrightarrow i + \overrightarrow j + 2\overrightarrow k $ and $\overrightarrow i + 2\overrightarrow j + \overrightarrow k $ and perpendicular to $\overrightarrow i + \overrightarrow j + \overrightarrow k $ is ...........
Correct Answer: $${{\widehat j - \widehat k} \over {\sqrt 2 }}$$ or $${{-\widehat j + \widehat k} \over {\sqrt 2 }}$$
1991
JEE Advanced
Numerical
IIT-JEE 1991
Given that $\overrightarrow a = \left( {1,1,1} \right),\,\,\overrightarrow c = \left( {0,1, - 1} \right),\,\overrightarrow a .\overrightarrow b = 3$ and $\overrightarrow a \times \overrightarrow b = \overrightarrow c ,$ then $\overrightarrow b \, = $.........
Correct Answer: $${{5\widehat i + 2\widehat j + 2\widehat k} \over 3}$$
1988
JEE Advanced
Numerical
IIT-JEE 1988
The components of a vector $\overrightarrow a $ along and perpendicular to a non-zero vector $\overrightarrow b $ are ......and .....respectively.
Correct Answer: $$\,\left( {{{\overrightarrow {a\,} .\,\overrightarrow b } \over {{{\left| {\overrightarrow b } \right|}^2}}}} \right)\overrightarrow b \,.\,,\,\,\overrightarrow a - \left( {{{\overrightarrow a \,.\,\overrightarrow b } \over {\left| {\overrightarrow b } \right|}}} \right)\overrightarrow b $$
1987
JEE Advanced
Numerical
IIT-JEE 1987
If the vectors $a\widehat i + \widehat j + \widehat k,\,\,\widehat i + b\widehat j + \widehat k$ and $\widehat i + \widehat j + c\widehat k$
$\left( {a \ne b \ne c \ne 1} \right)$ are coplannar, then the value of ${1 \over {\left( {1 - a} \right)}} + {1 \over {\left( {1 - b} \right)}} + {1 \over {\left( {1 - c} \right)}} = ..........$
$\left( {a \ne b \ne c \ne 1} \right)$ are coplannar, then the value of ${1 \over {\left( {1 - a} \right)}} + {1 \over {\left( {1 - b} \right)}} + {1 \over {\left( {1 - c} \right)}} = ..........$
Correct Answer: $$1$$
1987
JEE Advanced
Numerical
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 same plane having projecttions $1$ and $2$ along $\overrightarrow b $ and $\overrightarrow c, $ respectively, are given by ...........
Correct Answer: $$2\widehat i - \widehat j$$
1985
JEE Advanced
Numerical
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 the equations $\overrightarrow A \times \overrightarrow B = \overrightarrow {\,C} $ and $\overrightarrow A .\overrightarrow B = \overrightarrow {3\,} $ ..........
Correct Answer: $${5 \over 3}\widehat i + {2 \over 3}\widehat j + {2 \over 3}\widehat k$$
1985
JEE Advanced
Numerical
IIT-JEE 1985
If $\overrightarrow A \overrightarrow {\,B} \overrightarrow {\,C} $ are three non-coplannar vectors, then -
${{\overrightarrow A .\overrightarrow B \times \overrightarrow C } \over {\overrightarrow C \times \overrightarrow A .\overrightarrow B }} + {{\overrightarrow B .\overrightarrow A \times \overrightarrow C } \over {\overrightarrow C .\overrightarrow A \times \overrightarrow B }} = $ ................
${{\overrightarrow A .\overrightarrow B \times \overrightarrow C } \over {\overrightarrow C \times \overrightarrow A .\overrightarrow B }} + {{\overrightarrow B .\overrightarrow A \times \overrightarrow C } \over {\overrightarrow C .\overrightarrow A \times \overrightarrow B }} = $ ................
Correct Answer: $$0$$
1984
JEE Advanced
Numerical
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
$$\left( {\overrightarrow a - \overrightarrow d } \right)\left( {\overrightarrow b - \overrightarrow c } \right) = \left( {\overrightarrow b - \overrightarrow d } \right)\left( {\overrightarrow c - \overrightarrow a } \right) = 0$$
The point $D,$ then, is the ................ of the triangle $ABC.$
Correct Answer: Orthocenter
1981
JEE Advanced
Numerical
IIT-JEE 1981
Let $\overrightarrow A ,\overrightarrow B ,\overrightarrow C $ be vectors of length $3, 4, 5$ respectively. Let $\overrightarrow A $ be perpendicular to $\overrightarrow B + \overrightarrow C ,\overrightarrow B $ to $\overrightarrow C + \overrightarrow A $ to $\overrightarrow A + \overrightarrow B .$ Then the length of vector $\overrightarrow A + \overrightarrow B + \overrightarrow C $ is ..........
Correct Answer: $$5\sqrt 2 $$
1989
JEE Advanced
MCQ
IIT-JEE 1989
For any three vectors ${\overrightarrow a ,\,\overrightarrow b ,}$ and ${\overrightarrow c ,}$
$\left( {\overrightarrow a - \overrightarrow b } \right)\,.\,\left( {\overrightarrow b - \overrightarrow c } \right)\, \times \,\left( {\overrightarrow c - \overrightarrow a } \right)\, = \,2\overrightarrow {a\,} .\,\overrightarrow {b\,} \times \,\overrightarrow c .$
$\left( {\overrightarrow a - \overrightarrow b } \right)\,.\,\left( {\overrightarrow b - \overrightarrow c } \right)\, \times \,\left( {\overrightarrow c - \overrightarrow a } \right)\, = \,2\overrightarrow {a\,} .\,\overrightarrow {b\,} \times \,\overrightarrow c .$
A.
TRUE
B.
FALSE
1984
JEE Advanced
MCQ
IIT-JEE 1984
The points with position vectors $a+b,$ $a-b,$ and $a+kb$ are collinear for all real values of $k.$
A.
TRUE
B.
FALSE
1983
JEE Advanced
MCQ
IIT-JEE 1983
If $X.A=0, X.B=0, X.C=0$ for some non-zero vector $X,$ then $\left[ {A\,B\,C} \right] = 0$
A.
TRUE
B.
FALSE
1981
JEE Advanced
MCQ
IIT-JEE 1981
Let $\overrightarrow A ,\overrightarrow B $ and ${\overrightarrow C }$ be unit vectors suppose that $\overrightarrow A .\overrightarrow B = \overrightarrow A .\overrightarrow C = 0,$ and thatthe angle between ${\overrightarrow B }$ and ${\overrightarrow C }$ is $\pi /6.$ Then $\overrightarrow A = \pm 2\left( {\overrightarrow B \times \overrightarrow C } \right).$
A.
TRUE
B.
FALSE