Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Vector Algebra
0 votes

Show that the area of the parallelogram whose diagonals are given by $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\mid \overrightarrow{a} \times \overrightarrow{b}\mid}{2}$.Also find the area of the parallelogram whose diagonals are $2\hat i+\hat j+\hat k$ and $\hat i+3\hat j+\hat k$.

Can you answer this question?

1 Answer

0 votes
  • The area of a parallelogram whose diagonals are $\overrightarrow a$ and $\overrightarrow b$ is $\large\frac{1}{2}$$ |\overrightarrow a \times \overrightarrow b|$
Step 1:
Let $ABCD$ be a parallelogram with $A$ as the origin, let the position vectors of $B$ and $D$ be $\overrightarrow p$ and $\overrightarrow q$ respectively.Then,
$\overrightarrow {AB}= \overrightarrow p$ and $\overrightarrow {AD}= \overrightarrow q$
But we know $\overrightarrow {BC}= \overrightarrow {AD}$
Therefore $\overrightarrow {BC}= \overrightarrow q$
By triangle law of vectors, we have
$\overrightarrow {AB}+ \overrightarrow {BC}=\overrightarrow {AC}$
$\overrightarrow {AB}+ \overrightarrow {AD}=\overrightarrow {AC}$
Therefore $\overrightarrow p+ \overrightarrow q=\overrightarrow {AC}$-----(1)
Now let $\overrightarrow {AC}= \overrightarrow a$ and $\overrightarrow {BD}=\overrightarrow b,$ be the diagonals of the parallelogram $ABCD$
Then from equ(1) we have
$\overrightarrow p+ \overrightarrow q=\overrightarrow a$------(2)
and $\overrightarrow {BD}= \overrightarrow {OD}-\overrightarrow {OB}$
$=>\overrightarrow b= \overrightarrow q-\overrightarrow p$-----(3)
Add equ (1) and equ (2)
$\overrightarrow {2q}= \overrightarrow {a}+\overrightarrow {b}$
$=>\overrightarrow {q}=\large\frac{1}{2}$$(\overrightarrow {a}+\overrightarrow {b})$
Step 2:
Subtract equ(2) and equ(3) we get,
$2\overrightarrow p= \overrightarrow a-\overrightarrow b$
$=>\overrightarrow p= \large\frac{1}{2}$$(\overrightarrow {a}-\overrightarrow {b})$
Therefore $\overrightarrow p \times \overrightarrow q=\large\frac{1}{2}$$(\overrightarrow a-\overrightarrow b) \times \large\frac{1}{2}(\overrightarrow a+\overrightarrow b) $
$\qquad\qquad\qquad=\large\frac{1}{4} $$[(\overrightarrow a-\overrightarrow b) \times (\overrightarrow a+\overrightarrow b)]$
On expanding we get,
$\overrightarrow p \times \overrightarrow q=\large\frac{1}{4} $$\bigg[\overrightarrow a \times \overrightarrow a+ \overrightarrow a \times \overrightarrow b-\overrightarrow b \times \overrightarrow a-\overrightarrow b \times \overrightarrow b\bigg]$
But $\overrightarrow a \times \overrightarrow a=\overrightarrow b \times \overrightarrow b=0$ and $-\overrightarrow b \times \overrightarrow a=\overrightarrow a \times \overrightarrow b$
Therefore $\overrightarrow p \times \overrightarrow q=\large\frac{1}{4} $$\bigg[\overrightarrow a \times \overrightarrow b+ \overrightarrow a \times \overrightarrow b\bigg]$
$\quad\qquad\qquad\qquad=\large\frac{1}{2} $$\bigg[\overrightarrow a \times \overrightarrow b\bigg]$
So area of the paralleogram $ABCD$
$=|\overrightarrow p \times \overrightarrow q|=\large\frac{1}{2} $$|\overrightarrow a \times \overrightarrow b|$
Step 3:
Let $\overrightarrow a=2 \hat i+\hat j+\hat k$ and $\overrightarrow b=\hat i+3 \hat j+\hat k$
Hence the area of the parallelogram is
$\large\frac{1}{2} $$|\overrightarrow a \times \overrightarrow b|$
$=>\large\frac{1}{2} $$|\overrightarrow a \times \overrightarrow b|=\large\frac{1}{2}$$|(2 \hat i+\hat j+\hat k) \times (\hat i+3 \hat j+\hat k)|$
Let us determine $\overrightarrow a \times \overrightarrow b$
$\overrightarrow a \times \overrightarrow b=\begin {vmatrix} \hat i & \hat j & \hat k \\ 2 & 1 & 1 \\ 1 & 3 & 1 \end {vmatrix}$
$\quad\qquad=\hat i(1-3)-\hat j(2-1)+\hat k (6-1)$
$\quad\qquad=-2 \hat i-\hat j+5 \hat k $
$|\overrightarrow a \times \overrightarrow b|=\sqrt {(-2)^2+(-1)^2+(5)^2}$
$\qquad\qquad=\sqrt {4+1+25}$
$\qquad\qquad=\sqrt {30}$
Hence the area of the parallogram is $\large\frac{1}{2}$$\sqrt {30} sq.units$
answered May 31, 2013 by meena.p
edited May 31, 2013 by meena.p

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, AIPMT Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App