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