Ask Questions, Get Answers

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

Using vectors,find the area of the triangle ABC with vertices $A(1,2,3),B(2,-1,4)$ and $C(4,5,-1).$

Can you answer this question?

1 Answer

0 votes
  • Area of a triangle =$ \large\frac{1}{2}$$ | \overrightarrow{AB} \times \overrightarrow{BC}| $
Given $\overrightarrow {OA}=\hat i+2\hat j+3\hat k$, $\overrightarrow {OB}=2\hat i-\hat j+4\hat k$, and$\overrightarrow {OC}=4\hat i+5\hat j-\hat k$,
Area of a triangle is given by =$ \large\frac{1}{2}$$ | \overrightarrow{AB} \times \overrightarrow{BC}| $
Now let us determine $\overrightarrow{AB}$
$ \overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$
$\qquad=(2\hat i-\hat j+4\hat k)-(\hat i+2\hat j+3\hat k)$
$\qquad=\hat i-3\hat j+\hat k$
Now let us determine $\overrightarrow{AC}$
$ \overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}$
$\qquad=(4\hat i+5\hat j - \hat k)-(\hat i+2\hat j+3\hat k)$
$\qquad=3\hat i+3\hat j-4\hat k$
Now let us determine $\overrightarrow{AB} \times \overrightarrow{AC}$
$ \overrightarrow{AB} \times \overrightarrow{AC}=\begin{vmatrix} \hat i & \hat j & \hat k \\ 1 & -3 & 1 \\ 3 & 3 & -4 \end{vmatrix}$
$\qquad\qquad =\hat i(12-3)-\hat j(-4-3)+\hat k(3+9)$
$\qquad\qquad=9\hat i+7\hat j+12\hat k$
$ | \overrightarrow{AB} \times \overrightarrow{AC}| =\sqrt{9^2+7^2+12^2}$
Hence the area of the triangle is $\large \frac{\sqrt{274}}{2}$ sq. units
answered May 28, 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