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Please prove that, The vector area of a triangle whose sides are abar,bbar,cbar is 1/6(bxc+cxa+axb)
Dear student You are wrong here as it should be ½ in stead of 1/6 Let A be the endpoint of , B be the endpoint of vector , and C be the endpoint of vector .Then the vector from A to B is , and the vector from A to C is .So (1/2) | X| is the area of the triangle. ( magnitude of the cross-product is equal to the area of the parallelogram determined by the two vectors, and the area of the triangle is one-half the area of the parallelogram.)(B-A) X(C-A) = B X C - B X A - A X C + A X AThe cross product of a vector with itself is zero, and A X B = – B X A, so(B-A) X (C-A) = B X C + A X B + C X Awhich means that(1/2) | (B-A) X (C-A) | = (1/2) | B X C + A X B + C X A | = area of the triangle.
Let A be the endpoint of , B be the endpoint of vector , and C be the endpoint of vector .
Then the vector from A to B is , and the vector from A to C is .
So (1/2) | X| is the area of the triangle. ( magnitude of the cross-product is equal to the area of the parallelogram determined by the two vectors, and the area of the triangle is one-half the area of the parallelogram.)
(B-A) X(C-A) = B X C - B X A - A X C + A X A
The cross product of a vector with itself is zero, and A X B = – B X A, so(B-A) X (C-A) = B X C + A X B + C X A
which means that(1/2) | (B-A) X (C-A) | = (1/2) | B X C + A X B + C X A | = area of the triangle.
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