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Question
Using vectors find the area of triangle ABC with vertices A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).
Solution
The vertices of the triangle ABC are A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).
`vec(AB) = vec(OB) - vec(OA) = hati- 3hatj + hatk`
`vec(AC) = vec(OC) - vec(OA) = 3hati + 3hatj - 4hatk`
Area of the ∆ABC
`=1/2 |vec(AB) xx vec(AC)|`
`= 1/2|(hati,hatj,hatk),(1,-3,1),(3,3,-4)|`
`= 1/2|9hati + 7hatj +12hatk|`
`=1/2 sqrt(81+49+144)`
= `sqrt(274)/2`square units
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