Exam

JEE Advanced

Year

2013

Paper

Paper 1

Subject

Mathematics

Chapter

Vector Algebra

JEE Advanced 2013 · Paper 1 · Mathematics · Vector Algebra

Let → PR = 3 ^ i + ^ j - 2 ^ k and → SQ = ^ i - 3 ^ j - 4 ^ k...

Let \overset{\to}{\text{PR}} = 3 \overset{^}{i } + \overset{^}{j } - 2 \overset{^}{k} \text{and} \overset{\to}{\text{SQ}} = \overset{^}{i } - 3 \overset{^}{j } - 4 \overset{^}{k} determine diagonals of a parallelogram PQRS and \overset{\to}{\text{PT}} = \overset{^}{i } + 2 \overset{^}{j } + 3 \overset{^}{k} be another vector. Then the volume of the parallelepiped determined by the vectors $\overset{\to}{\text{PT}} \text{,} \overset{\to}{\text{PQ}} \text{and} \overset{\to}{\text{PS}}$ is

Step-by-step solution

$\overset{\to}{\text{PR}} = \overset{\to}{\text{PQ}} + \overset{\to}{\text{PS}}$
$\overset{\to}{\text{SQ}} = \overset{\to}{\text{PQ}} - \overset{\to}{\text{PS}}$
$\overset{\to}{\text{PS}} = \dfrac{\overset{\to}{\text{PR}} - \overset{\to}{\text{SQ}}}{2}$

$\overset{\to}{\text{PQ}} = \dfrac{\overset{\to}{\text{PR}} + \overset{\to}{\text{SQ}}}{2}$
$\text{V} = \left|\left[\overset{\to}{\text{PQ}} \overset{\to}{\text{PS}} \overset{\to}{\text{PT}}\right]\right|$
$\text{V} = \dfrac{1}{4} \left|\left[\overset{\to}{\text{PR}} + \overset{\to}{\text{SQ}} \text{,} \overset{\to}{\text{PR}} - \overset{\to}{\text{SQ}} \text{,} \overset{\to}{\text{PT}}\right]\right|$
$\text{V} = \dfrac{1}{2} \left|\left[\overset{\to}{\text{PR}} \text{,} \overset{\to}{\text{SQ}} \text{,} \overset{\to}{\text{PT}}\right]\right|$
V = \dfrac{1}{2}\left|\begin{cases}3 & 1 & - 2 \\ 1 & - 3 & - 4 \\ 1 & 2 & 3\end{cases}\right|
$V = \dfrac{1}{2}\left(- 3 - 7 - 10\right) = 10$