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Byju's Answer

Standard XII

Physics

Cross Product

Show that the...

Question

Show that the area of the triangle contained between the vectors a and b is one half of the magnitude of a × b.

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Solution

The following figure shows two vectors a and b at an angle θ extended to form a parallelogram.

From the above figure in ΔOMN,

sinθ= MN OM = MN | b | MN=| b |sinθ (1)

The magnitude for cross product of aand b is,

| a×b |=| a || b |sinθ

From the equation (1),

| a×b |=( OK )( MN )

Multiply and divide by 2.

| a×b |=( OK )( MN )× 2 2 =2×Area of ΔOMK Area of ΔOMK= 1 2 | a×b |

Thus, the area of the triangle contained between the vectors a and b is one half the magnitude of the a×b.

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