Explain sine law of vectors with the proof...
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Suppose a, b and c represent the sides of a triangle ABC in magnitude and direction.
Then we have a+b+c=0. Taking cross product with vector a we have a x a + a x b + a x c = 0. So a x b = c x a. Similarly, b x c = c x a.
Hence a x b = b x c = c x a
Therefore, |a x b| = |b x c| = |c x a|
So, ab sin C = bc sin A = ca sin B
Divide throughout by abc and take reciprocals
We get a/sin A = b/ sin B = c/ sin C which is the sine rule in a triangle.
Then we have a+b+c=0. Taking cross product with vector a we have a x a + a x b + a x c = 0. So a x b = c x a. Similarly, b x c = c x a.
Hence a x b = b x c = c x a
Therefore, |a x b| = |b x c| = |c x a|
So, ab sin C = bc sin A = ca sin B
Divide throughout by abc and take reciprocals
We get a/sin A = b/ sin B = c/ sin C which is the sine rule in a triangle.
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