← Back to Mathematics

Vector Algebra

619 Questions
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET
2003 JEE Mains MCQ
AIEEE 2003
If $\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $ then $\overrightarrow a + \overrightarrow b + \overrightarrow c = $
A.
$abc$
B.
$-1$
C.
$0$
D.
$2$
2003 JEE Advanced MCQ
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 $a\widehat i + \widehat k$ becomes minimum is
A.
$-3$
B.
$3$
C.
$1/\sqrt 3 $
D.
$\sqrt 3 $
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 $\,\left[ {\overrightarrow x \times \overrightarrow y \,\,\overrightarrow y \times \overrightarrow z \,\,\overrightarrow z \times \overrightarrow x } \right] = {1 \over {16}}{\left[ {\overrightarrow u \,\,\overrightarrow v \,\,\overrightarrow w } \right]^2}\,{\sec ^2}{\alpha \over 2}{\sec ^2}{\beta \over 2}{\sec ^2}{\gamma \over 2}.$
2002 JEE Mains MCQ
AIEEE 2002
If $\overrightarrow a \,\,,\,\,\overrightarrow b \,\,,\,\,\overrightarrow c $ are vectors such that $\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right] = 4$ then $\left[ {\overrightarrow a \, \times \overrightarrow b \,\,\overrightarrow b \times \,\overrightarrow c \,\,\overrightarrow c \, \times \overrightarrow a } \right] = $
A.
$16$
B.
$64$
C.
$4$
D.
$8$
2002 JEE Mains MCQ
AIEEE 2002
If the vectors $\overrightarrow c ,\overrightarrow a = x\widehat i + y\widehat j + z\widehat k$ and $\widehat b = \widehat j$ are such that $\overrightarrow a ,\overrightarrow c $ and $\overrightarrow b $ form a right handed system then ${\overrightarrow c }$ is :
A.
$z\widehat i - x\widehat k$
B.
$\overrightarrow 0 $
C.
$y\widehat j$
D.
$ - z\widehat i + x\widehat k$
2002 JEE Mains MCQ
AIEEE 2002
If the vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ from the sides $B C, C A$ and $A B$ respectively of a triangle $A B C$, then :
A.
$\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{b}}=0$
B.
$\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}$
C.
$\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}=0$
D.
$\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}=\overrightarrow{\mathbf{0}}$
2002 JEE Mains MCQ
AIEEE 2002
If $\left| {\overrightarrow a } \right| = 4,\left| {\overrightarrow b } \right| = 2$ and the angle between ${\overrightarrow a }$ and ${\overrightarrow b }$ is $\pi /6$ then ${\left( {\overrightarrow a \times \overrightarrow b } \right)^2}$ is equal to :
A.
$48$
B.
$16$
C.
$\overrightarrow a $
D.
none of these
2002 JEE Mains MCQ
AIEEE 2002
$\overrightarrow a = 3\widehat i - 5\widehat j$ and $\overrightarrow b = 6\widehat i + 3\widehat j$ are two vectors and $\overrightarrow c $ is a vector such that $\overrightarrow c = \overrightarrow a \times \overrightarrow b $ then $\left| {\overrightarrow a } \right|:\left| {\overrightarrow b } \right|:\left| {\overrightarrow c } \right|$ =
A.
$\sqrt {34} :\sqrt {45} :\sqrt {39} $
B.
$\sqrt {34} :\sqrt {45} :39$
C.
$34:39:45$
D.
$\,39:35:34$
2002 JEE Mains MCQ
AIEEE 2002
If $\left| {\overrightarrow a } \right| = 5,\left| {\overrightarrow b } \right| = 4,\left| {\overrightarrow c } \right| = 3$ thus what will be the value of $\left| {\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarrow c + \overrightarrow c .\overrightarrow a } \right|,$ given that $\overrightarrow a + \overrightarrow b + \overrightarrow c = 0$ :
A.
$25$
B.
$50$
C.
$-25$
D.
$-50$
2002 JEE Advanced MCQ
IIT-JEE 2002 Screening
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 JEE Advanced MCQ
IIT-JEE 2002 Screening
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 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 $V \le {L^3}\,\,.$
2001 JEE Advanced MCQ
IIT-JEE 2001 Screening
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 JEE Advanced MCQ
IIT-JEE 2001 Screening
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 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.
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$
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 $t.$
2000 JEE Advanced MCQ
IIT-JEE 2000 Screening
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 JEE Advanced MCQ
IIT-JEE 2000 Screening
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 JEE Advanced MCQ
IIT-JEE 2000 Screening
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 JEE Advanced MCQ
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 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 JEE Advanced MCQ
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 $c =$
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 JEE Advanced MSQ
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
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 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 $v .$
1998 JEE Advanced MCQ
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 $\left| c \right| = \sqrt 3 ,$ 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 JEE Advanced MCQ
IIT-JEE 1998
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 JEE Advanced MSQ
IIT-JEE 1998
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 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) $\left( {1 + {{\left| u \right|}^2}} \right)\left( {1 + {{\left| v \right|}^2}} \right) = {\left( {1 - u.v} \right)^2} + {\left| {u + v + \left( {u \times v} \right)} \right|^2}.$
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.)
1997 JEE Advanced Numerical
IIT-JEE 1997
If $A,B$ and $C$ are vectors such that $\left| B \right| = \left| C \right|.$ Prove that
$\left[ {\left( {A + B} \right) \times \left( {A + C} \right)} \right] \times \left( {B \times C} \right)\left( {B + C} \right) = 0\,\,.$
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 $k=$.........
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) = $ ..............
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 ................
1995 JEE Advanced MCQ
IIT-JEE 1995 Screening
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 JEE Advanced MCQ
IIT-JEE 1995 Screening
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 JEE Advanced MCQ
IIT-JEE 1995 Screening
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 JEE Advanced MCQ
IIT-JEE 1995 Screening
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 JEE Advanced MSQ
IIT-JEE 1994
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 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 $\overrightarrow a .$
1993 JEE Advanced MCQ
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 $c\widehat i + c\widehat j + b\widehat k$ 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 JEE Advanced MSQ
IIT-JEE 1993
Let $\vec a = 2\hat i - \hat j + \hat k,\vec b = \hat i + 2\hat j - \hat k$ and $\overrightarrow c = \widehat i + \widehat j - 2\widehat k - 2\widehat k$ 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 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.
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 ...........
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.
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 \, = $.........
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 $\overrightarrow R \,.\,\overrightarrow A = 0$
1989 JEE Advanced Numerical
IIT-JEE 1989
If vectors $\overrightarrow A ,\overrightarrow B ,\overrightarrow C $ are coplanar, show that $$\left| {\matrix{ {} & {\overrightarrow {a.} } & {} & {\overrightarrow {b.} } & {} & {\overrightarrow {c.} } \cr {\overrightarrow {a.} } & {\overrightarrow {a.} } & {\overrightarrow {a.} } & {\overrightarrow {b.} } & {\overrightarrow {a.} } & {\overrightarrow {c.} } \cr {\overrightarrow {b.} } & {\overrightarrow {a.} } & {\overrightarrow {b.} } & {\overrightarrow {b.} } & {\overrightarrow {b.} } & {\overrightarrow {c.} } \cr } } \right| = \overrightarrow 0 $$
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.
1989 JEE Advanced MCQ
IIT-JEE 1989
For any three vectors ${\overrightarrow a ,\,\overrightarrow b ,}$ and ${\overrightarrow c ,}$
$\left( {\overrightarrow a - \overrightarrow b } \right)\,.\,\left( {\overrightarrow b - \overrightarrow c } \right)\, \times \,\left( {\overrightarrow c - \overrightarrow a } \right)\, = \,2\overrightarrow {a\,} .\,\overrightarrow {b\,} \times \,\overrightarrow c .$
A.
TRUE
B.
FALSE
1988 JEE Advanced MCQ
IIT-JEE 1988
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.
$3$