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Vector Algebra

115 Questions Start JEE Advanced Test
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET BITSAT
2004 Q51 JEE Advanced MCQ
14 Mar 2026
If $\overrightarrow a = \left( {\widehat i + \widehat j + \widehat k} \right),\overrightarrow a .\overrightarrow b = 1$ and $\overrightarrow a \times \overrightarrow b = \widehat j - \widehat k,$ then $\overrightarrow b $ is
A.
$\widehat i - \widehat j + \widehat k$
B.
$2\widehat j - \widehat k$
C.
$\widehat i$
D.
$2\widehat i$
2004 Q52 JEE Advanced MCQ
14 Mar 2026
The unit vector which is orthogonal to the vector $3\overrightarrow i + 2\overrightarrow j + 6\overrightarrow k $ and is coplanar with the vectors $\,2\widehat i + \widehat j + \widehat k$ and $\,\widehat i - \widehat j + \widehat k$$\,\,\,$ is
A.
${{2\widehat i - 6\widehat j + \widehat k} \over {\sqrt {41} }}$
B.
${{2\widehat i - 3\widehat j} \over {\sqrt {13} }}$
C.
${{3\widehat i - \widehat k} \over {\sqrt {10} }}$
D.
${{4\widehat i + 3\widehat j - 3\widehat k} \over {\sqrt {34} }}$
2004 Q53 JEE Advanced Numerical
14 Mar 2026
If $\overrightarrow a ,\overrightarrow b ,\overrightarrow c $ and $\overrightarrow d $ are distinct vectors such that
$\,\overrightarrow a \times \overrightarrow c = \overrightarrow b \times \overrightarrow d $ and $\overrightarrow a \times \overrightarrow b = \overrightarrow c \times \overrightarrow d \,.$ Prove that
$\left( {\overrightarrow a - \overrightarrow d } \right).\left( {\overrightarrow b - \overrightarrow c } \right) \ne 0\,\,i.e.\,\,\,\overrightarrow a .\overrightarrow b + \overrightarrow d .\overrightarrow c \ne \overrightarrow d .\overrightarrow b + \overrightarrow a .\overrightarrow c $
2003 Q54 JEE Advanced MCQ
14 Mar 2026
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 i + \widehat k$ becomes minimum is
A.
$-3$
B.
$3$
C.
$1/\sqrt 3 $
D.
$\sqrt 3 $
2003 Q55 JEE Advanced Numerical
14 Mar 2026
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}.$
2002 Q56 JEE Advanced MCQ
14 Mar 2026
Let $\overrightarrow V = 2\overrightarrow i + \overrightarrow j - \overrightarrow k $ and $\overrightarrow W = \overrightarrow i + 3\overrightarrow k .$ If $\overrightarrow U $ is a unit vector, then the maximum value of the scalar triple product $\left| {\overrightarrow U \overrightarrow V \overrightarrow W } \right|$ is
A.
$-1$
B.
$\sqrt {10} + \sqrt 6 $
C.
$\sqrt {59} $
D.
$\sqrt {60} $
2002 Q57 JEE Advanced MCQ
14 Mar 2026
If ${\overrightarrow a }$ and ${\overrightarrow b }$ are two unit vectors such that ${\overrightarrow a + 2\overrightarrow b }$ and ${5\overrightarrow a - 4\overrightarrow b }$ are perpendicular to each other then the angle between $\overrightarrow a $ and $\overrightarrow b $ is
A.
${45^ \circ }$
B.
${60^ \circ }$
C.
${\cos ^{ - 1}}\left( {{1 \over 3}} \right)$
D.
${\cos ^{ - 1}}\left( {{2 \over 7}} \right)$
2002 Q58 JEE Advanced Numerical
14 Mar 2026
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}\,\,.$
2001 Q59 JEE Advanced MCQ
14 Mar 2026
If $\overrightarrow a \,,\,\overrightarrow b $ and $\overrightarrow c $ are unit vectors, then ${\left| {\overrightarrow a - \overrightarrow b } \right|^2} + {\left| {\overrightarrow b - \overrightarrow c } \right|^2} + {\left| {\overrightarrow c - \overrightarrow a } \right|^2}$ does NOT exceed
A.
$4$
B.
$9$
C.
$8$
D.
$6$
2001 Q60 JEE Advanced MCQ
14 Mar 2026
Let $\overrightarrow a = \overrightarrow i - \overrightarrow k ,\overrightarrow b = x\overrightarrow i + \overrightarrow j + \left( {1 - x} \right)\overrightarrow k $ and
$\overrightarrow c = y\overrightarrow i - x\overrightarrow j + \left( {1 + x - y} \right)\overrightarrow k .$ Then $\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$ depends on
A.
only $x$
B.
only $y$
C.
Neither $x$ Nor $y$
D.
both $x$ and $y$
2001 Q61 JEE Advanced Numerical
14 Mar 2026
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.
2001 Q62 JEE Advanced Numerical
14 Mar 2026
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$
2001 Q63 JEE Advanced Numerical
14 Mar 2026
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.$
2000 Q64 JEE Advanced MCQ
14 Mar 2026
If $\overrightarrow a \,,\,\overrightarrow b $ and $\overrightarrow c $ are unit coplanar vectors, then the scalar triple product $\left[ {2\overrightarrow a - \overrightarrow b ,2\overrightarrow b - \overrightarrow c ,2\overrightarrow c - \overrightarrow a } \right] = $
A.
$0$
B.
$1$
C.
$ - \sqrt 3 $
D.
$ \sqrt 3 $
2000 Q65 JEE Advanced MCQ
14 Mar 2026
If the vectors $\overrightarrow a ,\overrightarrow b $ and $\overrightarrow c $ form the sides $BC,$ $CA$ and $AB$ respectively of a triangle $ABC,$ then
A.
$\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarrow c + \overrightarrow c .\overrightarrow a = 0$
B.
$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $
C.
$\overrightarrow a .\overrightarrow b = \overrightarrow b .\overrightarrow c = \overrightarrow c .\overrightarrow a$
D.
$\overrightarrow a \times \overrightarrow b + \overrightarrow b \times \overrightarrow c + \overrightarrow c \times \overrightarrow a = \overrightarrow 0 $
2000 Q66 JEE Advanced MCQ
14 Mar 2026
Let the vectors $\overrightarrow a ,\overrightarrow b ,\overrightarrow c $ and $\overrightarrow d $ be such that
$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) = 0.$ Let ${P_1}$ and ${P_2}$ be planes determined
by the pairs of vectors $\overrightarrow a .\overrightarrow b $ and $\overrightarrow c .\overrightarrow d $ respectively. Then the angle between ${P_1}$ and ${P_2}$ is
A.
$0$
B.
${\pi \over 4}$
C.
${\pi \over 3}$
D.
${\pi \over 2}$
1999 Q67 JEE Advanced MCQ
14 Mar 2026
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 between $\left( {a \times b} \right)$ and $c$ is ${30^ \circ },$ then $\left| {\left( {a \times b} \right) \times c} \right| = $
A.
$2/3$
B.
$3/2$
C.
$2$
D.
$3$
1999 Q68 JEE Advanced MCQ
14 Mar 2026
Let $a=2i+j+k, b=i+2j-k$ and a unit vector $c$ be coplanar. If $c$ is perpendicular to $a,$ then $c =$
A.
${1 \over {\sqrt 2 }}\left( { - j + k} \right)$
B.
${1 \over {\sqrt 3 }}\left( {- i - j - k} \right)$
C.
${1 \over {\sqrt 5 }}\left( {i - 2j} \right)$
D.
${1 \over {\sqrt 3 }}\left( {i - j - k} \right)$
1999 Q69 JEE Advanced MSQ
14 Mar 2026
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
A.
$\left| u \right|$
B.
$\,\left| u \right| + \left| {u\,.\,a} \right|$
C.
$\,\left| u \right| + \left| {u\,.\,b} \right|$
D.
$\left| u \right| + u.\left( {a + b} \right)$
1999 Q70 JEE Advanced Numerical
14 Mar 2026
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 .$
1998 Q71 JEE Advanced MCQ
14 Mar 2026
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| = \sqrt 3 ,$ then
A.
$\alpha = 1,\,\,\beta = - 1$
B.
$\alpha = 1,\,\,\beta = \pm 1$
C.
$\alpha = - 1,\,\,\beta = \pm 1$
D.
$\alpha = \pm 1,\,\,\beta = 1$
1998 Q72 JEE Advanced MCQ
14 Mar 2026
For three vectors $u,v,w$ which of the following expression is not equal to any of the remaining three?
A.
$\,u \bullet \left( {v \times w} \right)$
B.
$\left( {v \times w} \right) \bullet u$
C.
$\,v \bullet \left( {u \times w} \right)$
D.
$\left( {u \times v} \right) \bullet w$
1998 Q73 JEE Advanced MSQ
14 Mar 2026
Which of the following expressions are meaningful?
A.
$u\left( {v \times w} \right)$
B.
$\left( {u \bullet v} \right) \bullet w$
C.
$\left( {u \bullet v} \right)w$
D.
$\,u\, \times \left( {v \bullet w} \right)$
1998 Q74 JEE Advanced Numerical
14 Mar 2026
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}.$
1998 Q75 JEE Advanced Numerical
14 Mar 2026
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.)
1997 Q76 JEE Advanced Numerical
14 Mar 2026
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\,\,.$
1997 Q77 JEE Advanced Numerical
14 Mar 2026
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=$.........
1996 Q78 JEE Advanced Numerical
14 Mar 2026
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) = $ ..............
1996 Q79 JEE Advanced Numerical
14 Mar 2026
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 ................
1995 Q80 JEE Advanced MCQ
14 Mar 2026
If $\overrightarrow a ,$ $\overrightarrow b $ and $\overrightarrow c $ are three non coplanar vectors, then
$\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right).\left[ {\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow c } \right)} \right]$ equals
A.
$0$
B.
$\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$
C.
$2\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$
D.
$-\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$
1995 Q81 JEE Advanced MCQ
14 Mar 2026
If $\overrightarrow a ,\overrightarrow b ,\overrightarrow c $ are non coplanar unit vectors such that $\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = {{\left( {\overrightarrow b + \overrightarrow c } \right)} \over {\sqrt 2 }},\,\,$ then the angle between $\overrightarrow a $ and $\overrightarrow b $ is
A.
${{3\pi } \over 4}$
B.
${{\pi } \over 4}$
C.
$\pi /2$
D.
$\pi $
1995 Q82 JEE Advanced MCQ
14 Mar 2026
Let $\overrightarrow a = \widehat i - \widehat j,\overrightarrow b = \widehat j - \widehat k,\overrightarrow c = \widehat k - \widehat i.$ If $\overrightarrow d $ is a unit vector such that $\overrightarrow a .\overrightarrow d = 0 = \left[ {\overrightarrow b \overrightarrow c \overrightarrow d } \right],$ then $\overrightarrow d $ equals
A.
$ \pm {{\widehat i + \widehat j - 2k} \over {\sqrt 6 }}$
B.
$ \pm {{\widehat i + \widehat j - k} \over {\sqrt 3 }}$
C.
$ \pm {{\widehat i + \widehat j + k} \over {\sqrt 3 }}$
D.
$ \pm \widehat k$
1995 Q83 JEE Advanced MCQ
14 Mar 2026
Let $\overrightarrow u ,\overrightarrow v $ and $\overrightarrow w $ be vectors such that $\overrightarrow u + \overrightarrow v + \overrightarrow w = 0.$ If $\left| {\overrightarrow u } \right| = 3,\left| {\overrightarrow v } \right| = 4$ and $\left| {\overrightarrow w } \right| = 5,$ then $\overrightarrow u .\overrightarrow v + \overrightarrow v .\overrightarrow w + \overrightarrow w .\overrightarrow u $ is
A.
$47$
B.
$-25$
C.
$0$
D.
$25$
1994 Q84 JEE Advanced MSQ
14 Mar 2026
The vector $\,{1 \over 3}\left( {2\widehat i - 2\widehat j + \widehat k} \right)$ is
A.
a unit vector
B.
makes an angle ${\pi \over 3}$ with the vector $\left( {2\widehat i - 4\widehat j + 3\widehat k} \right)$
C.
parallel to the vector $\left( { - \widehat i + \widehat j - {1 \over 2}\widehat k} \right)$
D.
perpendicular to the vector ${3\widehat i + 2\widehat j - 2\widehat k}$
1994 Q85 JEE Advanced Numerical
14 Mar 2026
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 .$
1993 Q86 JEE Advanced MCQ
14 Mar 2026
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 + c\widehat j + b\widehat k$ lie in a plane, then $c$ is
A.
the Arithmetic Mean of $a$ and $b$
B.
the Geometric Mean of $a$ and $b$
C.
the Harmonic Mean of $a$ and $b$
D.
equal to zero
1993 Q87 JEE Advanced MSQ
14 Mar 2026
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\widehat k$ be three vectors. A vector in the plane of ${\overrightarrow b }$ and ${\overrightarrow c }$, whose projection on ${\overrightarrow a }$ is of magnitude $\sqrt {2/3,} $ is :
A.
$2\widehat i + 3\widehat j - 3\widehat k$
B.
$2\widehat i + 3\widehat j + 3\widehat k$
C.
$-2\widehat i - \widehat j + 5\widehat k$
D.
$2\widehat i + \widehat j + 5\widehat k$
1993 Q88 JEE Advanced Numerical
14 Mar 2026
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.
1992 Q89 JEE Advanced Numerical
14 Mar 2026
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 ...........
1991 Q90 JEE Advanced Numerical
14 Mar 2026
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.
1991 Q91 JEE Advanced Numerical
14 Mar 2026
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 \, = $.........
1990 Q92 JEE Advanced Numerical
14 Mar 2026
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$
1989 Q93 JEE Advanced Numerical
14 Mar 2026
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 $$
1989 Q94 JEE Advanced Numerical
14 Mar 2026
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.
1989 Q95 JEE Advanced MCQ
14 Mar 2026
For any three vectors ${\overrightarrow a ,\,\overrightarrow b ,}$ and ${\overrightarrow c ,}$
$\left( {\overrightarrow a - \overrightarrow b } \right)\,.\,\left( {\overrightarrow b - \overrightarrow c } \right)\, \times \,\left( {\overrightarrow c - \overrightarrow a } \right)\, = \,2\overrightarrow {a\,} .\,\overrightarrow {b\,} \times \,\overrightarrow c .$
A.
TRUE
B.
FALSE
1988 Q96 JEE Advanced MCQ
14 Mar 2026
Let $\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,$ be three non-coplanar vectors and $\overrightarrow p ,\overrightarrow q ,\overrightarrow r,$ are vectors defined by the relations $\overrightarrow p = {{\overrightarrow b \times \overrightarrow c } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow q = {{\overrightarrow c \times \overrightarrow a } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow r = {{\overrightarrow a \times \overrightarrow b } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}}$ then the value of the expression $\left( {\overrightarrow a + \overrightarrow b } \right).\overrightarrow p + \left( {\overrightarrow b + \overrightarrow c } \right).\overrightarrow q + \left( {\overrightarrow c + \overrightarrow a } \right),\overrightarrow r $ is equal to
A.
$0$
B.
$1$
C.
$2$
D.
$3$
1988 Q97 JEE Advanced Numerical
14 Mar 2026
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.
1988 Q98 JEE Advanced Numerical
14 Mar 2026
The components of a vector $\overrightarrow a $ along and perpendicular to a non-zero vector $\overrightarrow b $ are ......and .....respectively.
1987 Q99 JEE Advanced MCQ
14 Mar 2026
The number of vectors of unit length perpendicular to vectors $\overrightarrow a = \left( {1,1,0} \right)$ and $\overrightarrow b = \left( {0,1,1} \right)$ is
A.
one
B.
two
C.
three
D.
infinite
1987 Q100 JEE Advanced Numerical
14 Mar 2026
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 $ABC$)