1
IIT-JEE 2009 Paper 1 Offline
+3
-1

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ and $$\overrightarrow d$$ are unit vectors such that $$(\overrightarrow a \times \overrightarrow b )\,.\,(\overrightarrow c \times \overrightarrow d ) = 1$$ and $$\overrightarrow a \,.\,\overrightarrow c = {1 \over 2}$$, then

A
$$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c$$ are non-coplanar
B
$$\overrightarrow b \,,\,\overrightarrow c ,\overrightarrow d$$ are non-coplanar
C
$$\overrightarrow b \,,\overrightarrow d$$ are non-parallel
D
$$\overrightarrow a ,\overrightarrow d$$ parallel and $$\overrightarrow b ,\overrightarrow c$$ are parallel
2
IIT-JEE 2008 Paper 2 Offline
+3
-1
Let two non-collinear unit vectors $$\widehat a$$ and $$\widehat b$$ form an acute angle. A point $$P$$ moves so that at any time $$t$$ the position vector $$\overrightarrow {OP}$$ (where $$O$$ is the origin) is given by $$\widehat a\cos t + \widehat b\sin t.$$ When $$P$$ is farthest from origin $$O,$$ let $$M$$ be the length of $$\overrightarrow {OP}$$ and $$\widehat u$$ be the unit vector along $$\overrightarrow {OP}$$. Then :
A
$$\widehat u = {{\widehat a + \widehat b} \over {\left| {\widehat a + \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + \widehat a.\,\widehat b} \right)^{1/2}}$$
B
$$\widehat u = {{\widehat a - \widehat b} \over {\left| {\widehat a - \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + \widehat a.\,\widehat b} \right)^{1/2}}$$
C
$$\widehat u = {{\widehat a + \widehat b} \over {\left| {\widehat a + \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + 2\widehat a.\,\widehat b} \right)^{1/2}}$$
D
$$\widehat u = {{\widehat a - \widehat b} \over {\left| {\widehat a - \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + 2\widehat a.\,\widehat b} \right)^{1/2}}$$
3
IIT-JEE 2008 Paper 2 Offline
+3
-1
Consider the lines

$${L_1}:{{x + 1} \over 3} = {{y + 2} \over 1} = {{z + 1} \over 2}$$

$${L_2}:{{x - 2} \over 1} = {{y + 2} \over 2} = {{z - 3} \over 3}$$

The unit vector perpendicular to both $${L_1}$$ and $${L_2}$$ is :

A
$${{ - \widehat i + 7\widehat j + 7\widehat k} \over {\sqrt {99} }}$$
B
$${{ - \widehat i - 7\widehat j + 5\widehat k} \over {5\sqrt 3 }}$$
C
$${{ - \widehat i + 7\widehat j + 5\widehat k} \over {5\sqrt 3 }}$$
D
$${{7\widehat i - 7\widehat j - \widehat k} \over {\sqrt {99} }}$$
4
IIT-JEE 2008 Paper 2 Offline
+3
-1

Consider the lines,

$${L_1}:{{x + 1} \over 3} = {{y + 2} \over 1} = {{z + 1} \over 2}$$

$${L_2}:{{x - 2} \over 1} = {{y - 2} \over 2} = {{z - 3} \over 3}$$

The shortest distance between $${L_1}$$ and $${L_2}$$ is :

A
$$0$$
B
$${17 \over {\sqrt 3 }}$$
C
$${41 \over {5\sqrt 3 }}$$
D
$${17 \over {5\sqrt 3 }}$$
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