Ncert Exercises & Solutions

Question 7. 4. Show that the area of the triangle contained between the vectors $\overset{⇀}{a} and \overset{⇀}{b}$ is one half of the magnitude of $\overset{⇀}{a} \times \overset{⇀}{b}$ .

Consider two vectors

\vec{OK} = |\vec{a}|

and

\vec{OM} = |\vec{b}|

, inclined at an angle θ, as shown in the following figure.

In ΔOMN, we can write the relation:

sinθ = \frac{MN}{OM} = \frac{MN}{|\vec{b}|}

MN = |\vec{b}| sinθ

|\vec{a} \times \vec{a}| = |\vec{a}| |\vec{b}| sinθ

= OK \cdot MN \times \frac{2}{2}

= 2 \times Area of △ OMK

∴ Area of △ OMK = \frac{1}{2} |\vec{a} \times \vec{b}|