1.
Let $$P,\,Q,\,R$$ and $$S$$ be the points on the plane with position vectors $$ - 2\hat i - \hat j,\,4\hat i,\,3\hat i + 3\hat j$$ and $$ - 3\hat i + 2\hat j$$ respectively, The quadrilateral $$PQRS$$ must be a :
A.
parallelogram, which is neither a rhombus nor a rectangle
B.
square
C.
rectangle, but not a square
D.
rhombus, but not a square
Answer :
parallelogram, which is neither a rhombus nor a rectangle
Let $$P\left( {{x_1},\,{y_1},\,{z_1}} \right)$$ and $$Q\left( {{x_2},\,{y_2},\,{z_2}} \right)$$ be the initial and final points of the vector whose projections on the three coordinate axes are $$6,\, - 3,\,2$$
then
$${x_2} - {x_1} = 6\,;\,\,\,{y_2} - {y_1} = - 3\,;\,\,\,{z_2} - \,{z_1} = 2$$
So that direction ratios of $$\overrightarrow {PQ} $$ are $$6,\, - 3,\,2$$
$$\therefore $$ Direction cosines of $$\overrightarrow {PQ} $$ are
$$\eqalign{
& \frac{6}{{\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }},\,\frac{{ - 3}}{{\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }},\,\frac{2}{{\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }} \cr
& = \frac{6}{7},\,\frac{{ - 3}}{7},\,\frac{2}{7} \cr} $$
3.
If $$\overrightarrow a $$ and $$\overrightarrow b $$ are two vectors of magnitude $$2$$ inclined at an angle $${60^ \circ }$$ then the
angle between $$\overrightarrow a $$ and $$\overrightarrow a + \overrightarrow b $$ is :
Here, $$\left| {\overrightarrow a } \right| = 2 = \left| {\overrightarrow b } \right|$$ and $$\overrightarrow a .\overrightarrow b = 2.2\cos \,{60^ \circ } = 2$$
If the angle between $$\overrightarrow a $$ and $$\left( {\overrightarrow a + \overrightarrow b } \right)$$ be $$\theta $$ then
$$\eqalign{
& \,\,\,\,\,\,\,\,\,\,\overrightarrow a .\left( {\overrightarrow a + \overrightarrow b } \right) = \left| {\overrightarrow a } \right|\,\,\left| {\overrightarrow a + \overrightarrow b } \right|\cos \,\theta \cr
& {\text{or }}{\left| {\overrightarrow a } \right|^2}\, + \overrightarrow a .\overrightarrow b = 2\,\left| {\overrightarrow a + \overrightarrow b } \right|\cos \,\theta \cr
& {\text{or }}4 + 2 = 2\left| {\overrightarrow a + \overrightarrow b } \right|\cos \,\theta \cr
& \therefore \,\cos \,\theta = \frac{3}{{\left| {\overrightarrow a + \overrightarrow b } \right|}} \cr
& {\text{Now, }}{\left| {\overrightarrow a + \overrightarrow b } \right|^2} = {\left( {\overrightarrow a + \overrightarrow b } \right)^2} \cr
& = {\overrightarrow a ^2} + {\overrightarrow b ^2} + 2\overrightarrow a .\overrightarrow b \cr
& = 4 + 4 + 2.2 \cr
& = 12 \cr
& \therefore \,\,\,\left| {\overrightarrow a + \overrightarrow b } \right| = 2\sqrt 3 \cr
& \therefore \,\,\cos \,\theta = \frac{3}{{2\sqrt 3 }} = \frac{{\sqrt 3 }}{2}\,\,\,\,\, \Rightarrow \theta = {30^ \circ } \cr} $$
4.
Let $$\overrightarrow a = 2\overrightarrow i - \overrightarrow j + \overrightarrow k ,\,\overrightarrow b = \overrightarrow i + 2\overrightarrow j - \overrightarrow k $$ and $$\overrightarrow c = \overrightarrow i + \overrightarrow j - 2\overrightarrow k .$$ A vector in the plane of $$\overrightarrow b $$ and $$\overrightarrow c $$ whose projection on $$\overrightarrow a $$ has the magnitude $$\sqrt {\frac{2}{3}} $$ is :
A.
$$2\overrightarrow i + 3\overrightarrow j - 3\overrightarrow k $$
B.
$$2\overrightarrow i + 3\overrightarrow j + 3\overrightarrow k $$
C.
$$ - 2\overrightarrow i - \overrightarrow j + 5\overrightarrow k $$
D.
$$2\overrightarrow i + \overrightarrow j + 5\overrightarrow k $$
Answer :
$$ - 2\overrightarrow i - \overrightarrow j + 5\overrightarrow k $$
6.
If \[\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c \] are noncoplanar nonzero vectors then \[\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow a \times \overrightarrow c } \right) + \left( {\overrightarrow b \times \overrightarrow c } \right) \times \left( {\overrightarrow b \times \overrightarrow a } \right) + \left( {\overrightarrow c \times \overrightarrow a } \right) \times \left( {\overrightarrow c \times \overrightarrow b } \right)\] is equal to :
A.
\[{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]^2}\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right)\]
B.
\[\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right)\]
C.
\[\overrightarrow 0 \]
D.
none of these
Answer :
\[\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right)\]
\[\begin{array}{l}
\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow a \times \overrightarrow c } \right) = \left( {\overrightarrow a \times \overrightarrow b .\overrightarrow c } \right)\overrightarrow a - \left( {\overrightarrow a \times \overrightarrow b .\overrightarrow a } \right)\overrightarrow c \\
= \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow a - \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow c \\
= \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow a \\
{\rm{Similarly,}}\,\left( {\overrightarrow b \times \overrightarrow c } \right) \times \left( {\overrightarrow b \times \overrightarrow a } \right) = \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right]\overrightarrow b \\
\left( {\overrightarrow c \times \overrightarrow a } \right) \times \left( {\overrightarrow c \times \overrightarrow b } \right) = \left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right]\overrightarrow c \\
\therefore {\rm{ the\, expression}} = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right)
\end{array}\]
7.
Let $$\vec p$$ and $$\vec q$$ be the position vectors of $$P$$ and $$Q$$ respectively, with respect to $$O$$ and $$\left| {\vec p} \right| = p,\,\left| {\vec q} \right| = q.$$ The points $$R$$ and $$S$$ divide $$PQ$$ internally and externally in the ratio 2 : 3 respectively. If $$OR$$ and $$OS$$ are perpendicular then :
8.
Let $$\vec \alpha = 3\hat i + \hat j$$ and $$\vec \beta = 2\hat i - \hat j + 3\hat k.$$
If $$\vec \beta = {{\vec \beta }_1} - {{\vec \beta }_2},$$ where $${{\vec \beta }_1}$$ is parallel to $${\vec \alpha }$$ and $${{\vec \beta }_2}$$ is perpendicular to $${\vec \alpha },$$ then $${{\vec \beta }_1} \times {{\vec \beta }_2}$$ is equal to :
A.
$$ - 3\hat i + 9\hat j + 5\hat k$$
B.
$$3\hat i - 9\hat j - 5\hat k$$
C.
$$\frac{1}{2}\left( { - 3\hat i + 9\hat j + 5\hat k} \right)$$
D.
$$\frac{1}{2}\left( {3\hat i - 9\hat j + 5\hat k} \right)$$
9.
Let $$A = \left( {1,\,2,\,2} \right),\,B = \left( {2,\,3,\,6} \right)$$ and $$C = \left( {3,\,4,\,12} \right).$$ The direction cosines of a line equally inclined with $$OA,\,OB$$ and $$OC$$ where $$O$$ is the origin, are :
A.
$$\frac{1}{{\sqrt 2 }},\,\frac{{ - 1}}{{\sqrt 2 }},\,0$$
B.
$$\frac{1}{{\sqrt 2 }},\,\frac{1}{{\sqrt 2 }},\,0$$
C.
$$\frac{1}{{\sqrt 3 }},\, - \frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }}$$
10.
Let $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$ be three unit vectors and $$\overrightarrow a .\overrightarrow b = \overrightarrow a .\overrightarrow c = 0.$$ If the angle between $$\overrightarrow b $$ and $$\overrightarrow c $$ is $$\frac{\pi }{3}$$ then $$\left| {\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]} \right|$$ is equal to :
$$\eqalign{
& \overrightarrow a \bot \overrightarrow b ,\,\overrightarrow a \bot \overrightarrow c ,\,{\text{i}}{\text{.e}}{\text{., }}\overrightarrow a ||\left( {\overrightarrow b \times \overrightarrow c } \right){\text{ and }}\left| {\overrightarrow b \times \overrightarrow c } \right| = \left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|\sin \frac{\pi }{3} = \frac{{\sqrt 3 }}{2} \cr
& \left| {\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]} \right| = \left| {\overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)} \right| = \left| {\overrightarrow a } \right|\left| {\overrightarrow b \times \overrightarrow c } \right|\cos \,{0^ \circ } = 1.\frac{{\sqrt 3 }}{2}.1 = \frac{{\sqrt 3 }}{2} \cr} $$