1
IIT-JEE 1986
+2
-0.5
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
$${3 \over 4}\left( {a_1^2 + a_2^2 + a_3^2} \right)\left( {b_1^2 + b_2^2 + b_3^2} \right)\left( {c_1^2 + c_2^2 + c_3^2} \right)$$
2
IIT-JEE 1983
+1
-0.25
The volume of the parallelopiped whose sides are given by
$$\overrightarrow {OA} = 2i - 2j,\,\overrightarrow {OB} = i + j - k,\,\overrightarrow {OC} = 3i - k,$$ is
A
$${4 \over {13}}$$
B
$$4$$
C
$${2 \over 7}$$
D
none of these
3
IIT-JEE 1983
+1
-0.25
The points with position vectors $$60i+3j,$$ $$40i-8j,$$ $$ai-52j$$ are collinear if
A
$$a=-40$$
B
$$a=40$$
C
$$a=20$$
D
none of these
4
IIT-JEE 1982
+2
-0.5
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
$$\overrightarrow a \,.\,\overrightarrow b = \overrightarrow b \,.\,\overrightarrow c = \overrightarrow c \,.\,\overrightarrow a = 0$$
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