Example 24 - Find area of a triangle having A(1, 1, 1), as - Examples

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Example 24 Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1) as its vertices. A (1, 1, 1) , B (1, 2, 3) ,C (2, 3, 1) The area of triangle ABC is 1﷮2﷯ 𝐴𝐵﷯× 𝐴𝐶﷯﷯ 𝑨𝑩﷯ × 𝑨𝑪﷯ = 𝑖﷯﷮ 𝑗﷯﷮ 𝑘﷯﷮0﷮1﷮2﷮1﷮2﷮0﷯﷯ = 𝑖﷯ 1×0﷯−(2×2)﷯ − 𝑗﷯ 0×0﷯−(1×2)﷯ + 𝑘﷯ 0×2﷯−(1×1)﷯ = −4 𝒊﷯ + 2 𝒋﷯ – 1 𝒌﷯ Magnitude of 𝐴𝐵﷯ × 𝐴𝐶﷯ = ﷮ −4﷯2+22+ −1﷯2﷯ 𝐴𝐵﷯ × 𝐴𝐶﷯﷯ = ﷮16+4+1﷯ = ﷮21﷯ So, area of triangle ABC = 1﷮2﷯ 𝐴𝐵﷯ × 𝐴𝐶﷯﷯ = 1﷮2﷯ × ﷮21﷯ = ﷮21﷯﷮2﷯ Therefore, the required area is ﷮𝟐𝟏﷯﷮𝟐﷯

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