19. Let $\overrightarrow { \mathrm { A } }$ be vector parallel to line of intersection of planes $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$ through origin. $\mathrm { P } _ { 1 }$ is parallel to the vectors $2 \hat { \mathrm { j } } + 3 \hat { \mathrm { k } }$ and $4 \hat { j } - 3 \hat { k }$ and $P _ { 2 }$ is parallel to $\hat { j } - \hat { k }$ and $3 \hat { i } + 3 \hat { j }$, then the angle between vector $\vec { A }$ and $2 \hat { i } + \hat { j } - 2 \hat { k }$ is (A) $\frac { \pi } { 2 }$ (B) $\frac { \pi } { 4 }$ (C) $\frac { \pi } { 6 }$ (D) $\frac { 3 \pi } { 4 }$ Sol. (B), (D) Vector AB is parallel to $[ ( 2 \hat { \mathrm { i } } + 3 \hat { \mathrm { k } } ) \times ( 4 ) - 3 \hat { \mathrm { k } } ] \times [ ( \hat { \mathrm { j } } - \hat { \mathrm { k } } ) \times ( 3 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } ) ] = 54 ( \hat { \mathrm { j } } - \hat { \mathrm { k } } )$ Let $\theta$ is the angle between the vector, then $$\cos \theta = \pm \left( \frac { 54 + 108 } { 3.54 \sqrt { 2 } } \right) = \pm \frac { 1 } { \sqrt { 2 } }$$ Hence $\theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 }$.
Let $\overrightarrow { \mathrm { A } }$ be vector parallel to line of intersection of planes $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$ through origin. $\mathrm { P } _ { 1 }$ is parallel to the vectors $2 \hat { \mathrm { j } } + 3 \hat { \mathrm { k } }$ and $4 \hat { j } - 3 \hat { k }$ and $P _ { 2 }$ is parallel to $\hat { j } - \hat { k }$ and $3 \hat { i } + 3 \hat { j }$, then the angle between vector $\vec { A }$ and $2 \hat { i } + \hat { j } - 2 \hat { k }$ is
19. Let $\overrightarrow { \mathrm { A } }$ be vector parallel to line of intersection of planes $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$ through origin. $\mathrm { P } _ { 1 }$ is parallel to the vectors $2 \hat { \mathrm { j } } + 3 \hat { \mathrm { k } }$ and $4 \hat { j } - 3 \hat { k }$ and $P _ { 2 }$ is parallel to $\hat { j } - \hat { k }$ and $3 \hat { i } + 3 \hat { j }$, then the angle between vector $\vec { A }$ and $2 \hat { i } + \hat { j } - 2 \hat { k }$ is\\
(A) $\frac { \pi } { 2 }$\\
(B) $\frac { \pi } { 4 }$\\
(C) $\frac { \pi } { 6 }$\\
(D) $\frac { 3 \pi } { 4 }$
Sol. (B), (D)\\
Vector AB is parallel to $[ ( 2 \hat { \mathrm { i } } + 3 \hat { \mathrm { k } } ) \times ( 4 ) - 3 \hat { \mathrm { k } } ] \times [ ( \hat { \mathrm { j } } - \hat { \mathrm { k } } ) \times ( 3 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } ) ] = 54 ( \hat { \mathrm { j } } - \hat { \mathrm { k } } )$\\
Let $\theta$ is the angle between the vector, then
$$\cos \theta = \pm \left( \frac { 54 + 108 } { 3.54 \sqrt { 2 } } \right) = \pm \frac { 1 } { \sqrt { 2 } }$$
Hence $\theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 }$.\\