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Applications of Cross Product

If a⃗ and b⃗ ...

Question

If $\to a$ and $\to b$ are the two sides of a triangle, then the area of triangle will be given by $| \to a \times \to b |$

A

True

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B

False

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Solution

The correct option is B False

As shown in the figure if $\to a$ and $\to b$ represent the two sides of a triangle then the area of triangle is given by $\frac{1}{2} | \to a \times \to b |$ and not $\to a \times \to b$
Why is it so?
Look area of a triangle is given by $\frac{1}{2}$ (Base) $\times$ ( Perpendicular distance of the base from the opposite vertex)
So $\frac{1}{2} | \to a \times \to b |$ = $\frac{1}{2} (| \to a | | \to b | s i n ϕ)$ where $ϕ$ is the angle between $\to a$ and $\to b$
Now $| \to b |$ sin $ϕ$ will be nothing but the perpendicular distance of $\to a$ from the opposite vertex)
So, $\frac{1}{2} (| \to a | | \to b | s i n ϕ)$ = $\frac{1}{2} (| \to a |)$ x perpendicular distance of $\to a$ from the opposite vertex)
Which is nothing but the area of the triangle.

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