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:

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

$MN = |\vec{b}| sinθ$

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

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

$= 2 \times Area of △ OMK$

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

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