logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
0 votes

The area of the parallelogram having a diagonal $\overrightarrow{3i}+\overrightarrow{j}-\overrightarrow{k}$ and a side $\overrightarrow{i}-\overrightarrow{3j}+\overrightarrow{4k} $ is

 \[\begin{array}{1 1}(1)\;10\sqrt{3} &(2)\;6\sqrt{30}\\(3)\;\frac{3}{2}\sqrt{30}&(4)\;3\sqrt{30}\end{array}\]

Can you answer this question?
 
 

1 Answer

0 votes
Let $\overrightarrow{AB}=\overrightarrow{i}-3\overrightarrow{j}+4 \overrightarrow{k}$
$\overrightarrow{AC}=3\overrightarrow{i} + \overrightarrow{j} - \overrightarrow{k}$
The vector area of the triangle ABC with adjacent sides $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is
$\qquad= \large\frac{1}{2}$$ \overrightarrow{AB} \times \overrightarrow{AC}$
$\qquad=\large \frac{1}{2} $$(\overrightarrow{i}-3 \overrightarrow{j} +4 \overrightarrow{k} ) \times 3(\overrightarrow{i} + \overrightarrow{j}- \overrightarrow{k})$
$\qquad= \large\frac{1}{2}$$ \begin{vmatrix} \overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k} \\1 & -3 & 4 \\3 & 1 & -1 \end{vmatrix}$
$\qquad= \large\frac{1}{2}$$ [ \overrightarrow{i} (3-4) -\overrightarrow{j}(-1-12)+\overrightarrow{k}(1+9)]$
$\qquad= \large\frac{1}{2}$$ [- \overrightarrow{i} +13 \overrightarrow{j}+10 \overrightarrow{k}]$
Area of triangle = $\large\frac{1}{2}$$ |- \overrightarrow{i} +13 \overrightarrow{j}+10 \overrightarrow{k}|$
$\qquad= \large\frac{1}{2} $$ \sqrt {(-1)^2+13^2+10^2}$
$\qquad= \large\frac{1}{2} $$ \sqrt {1+169+100}$
$\qquad= \large\frac{1}{2} $$ \sqrt {270}$
$\qquad= \large\frac{1}{2} $$ \sqrt {9 \times 30}$
$\qquad= \large\frac{3}{2} $$ \sqrt {30}$
Area of parallelogram $ ABCD $=2 $\times$ area of $\Delta ABC$
$\qquad= 2 \times \large\frac{3}{2} $$\sqrt {30} =3 \sqrt {30}$
Hence 4 is the correct answer
answered May 5, 2014 by meena.p
 

Related questions

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