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Vector Algebra

638 Questions Start JEE Mains Test
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET BITSAT
1996 Q601 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 Q602 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 Q603 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 Q604 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 Q605 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 Q606 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 Q607 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 Q608 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 Q609 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 Q610 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 Q611 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 Q612 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 Q613 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 Q614 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 Q615 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 Q616 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 Q617 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 Q618 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 Q619 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 Q620 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 Q621 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 Q622 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 Q623 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 $ABC$)
1987 Q624 JEE Advanced Numerical
14 Mar 2026
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 ${1 \over {\left( {1 - a} \right)}} + {1 \over {\left( {1 - b} \right)}} + {1 \over {\left( {1 - c} \right)}} = ..........$
1987 Q625 JEE Advanced Numerical
14 Mar 2026
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 ...........
1986 Q626 JEE Advanced MCQ
14 Mar 2026
Let $\overrightarrow a = {a_1}i + {a_2}j + {a_3}k,\,\,\,\overrightarrow b = {b_1}i + {b_2}j + {b_3}k$ and $\overrightarrow c = {c_1}i + {c_2}j + {c_3}k$ be three non-zero vectors such that $\overrightarrow c $ is a unit vector perpendicular to both the vectors $\overrightarrow a $ and $\overrightarrow b .$ If the angle between $\overrightarrow a $ and $\overrightarrow b $ is ${\pi \over 6},$ then
${\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{b_1}} & {{b_2}} & {{b_3}} \cr {{c_1}} & {{c_2}} & {{c_3}} \cr } } \right|^2}$ is equal to
A.
$0$
B.
$1$
C.
${1 \over 4}\left( {a_1^2 + a_2^2 + a_2^3} \right)\left( {b_1^2 + b_2^2 + b_3^2} \right)$
D.
${3 \over 4}\left( {a_1^2 + a_2^2 + a_3^2} \right)\left( {b_1^2 + b_2^2 + b_3^2} \right)\left( {c_1^2 + c_2^2 + c_3^2} \right)$
1986 Q627 JEE Advanced Numerical
14 Mar 2026
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 $\lambda .$
1985 Q628 JEE Advanced Numerical
14 Mar 2026
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\,} $ ..........
1985 Q629 JEE Advanced Numerical
14 Mar 2026
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 }} = $ ................
1984 Q630 JEE Advanced Numerical
14 Mar 2026
$A, B, C$ 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 $ABC.$

1984 Q631 JEE Advanced MCQ
14 Mar 2026
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 Q632 JEE Advanced MCQ
14 Mar 2026
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
1982 Q633 JEE Advanced MCQ
14 Mar 2026
For non-zero vectors ${\overrightarrow a ,\,\overrightarrow b ,\overrightarrow c },$ $\left| {\left( {\overrightarrow a \times \overrightarrow b } \right).\overrightarrow c } \right| = \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|$ holds if and only if
A.
$\overrightarrow a \,.\,\overrightarrow b = 0,\overrightarrow b \,.\,\overrightarrow c = 0$
B.
$\overrightarrow b \,.\,\overrightarrow c = 0,\overrightarrow c \,.\,\overrightarrow a = 0$
C.
$\overrightarrow c \,.\,\overrightarrow a = 0,\overrightarrow a \,.\,\overrightarrow b = 0$
D.
$\overrightarrow a \,.\,\overrightarrow b = \overrightarrow b \,.\,\overrightarrow c = \overrightarrow c \,.\,\overrightarrow a = 0$
1982 Q634 JEE Advanced Numerical
14 Mar 2026
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.
1982 Q635 JEE Advanced Numerical
14 Mar 2026
${A_1},{A_2},.................{A_n}$ 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)} $
1981 Q636 JEE Advanced MCQ
14 Mar 2026
The scalar $\overrightarrow A .\left( {\overrightarrow B + \overrightarrow C } \right) \times \left( {\overrightarrow A + \overrightarrow B + \overrightarrow C } \right)$ equals :
A.
$0$
B.
$\left[ {\overrightarrow A \,\overrightarrow B \,\overrightarrow C } \right] + \left[ {\overrightarrow B \,\overrightarrow C \,\overrightarrow A } \right]$
C.
$\left[ {\overrightarrow A \,\overrightarrow B \,\overrightarrow C } \right]$
D.
None of these
1981 Q637 JEE Advanced Numerical
14 Mar 2026
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 ..........
1981 Q638 JEE Advanced MCQ
14 Mar 2026
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