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Question
Find the area of the triangle whose vertices are A(3, – 1, 2), B(1, – 1, – 3) and C(4, – 3, 1)
Solution
The given vertices of the triangle ABC are A(3, – 1, 2), B(1, – 1, – 3) and C(4, – 3, 1)
`vec"OA" = 3hat"i" - hat"j" + 2hat"k"`
`vec"OB" = hat"i" - hat"j" - 3hat"k"`
`vec"OC" = 4hat"i" - 3hat"j" + hat"k"`
`vec"AB" = vec"OB" - vec"OA"`
= `(hat"i" - hat"j" - 3hat"k") - (3hat"i" - hat"j" + 2hat"k")`
= `hat"i" - hat"j" - 3hat"k" - 3hat"i" + hat"j" - 2hat"k"`
`vec"AB" = -2hat"i" - 5hat"k"`
`vec"AC" = vec"OC" - vec"OA"`
= `(4hat"i" - 3hat"j" + hat"k") - (3hat"i" - hat"j" + 2hat"k")`
= `4hat"i" - 3hat"j" + hat"k" - 3hat"i" + hat"j" - 2hat"k"`
`vec"AC" = hat"i" - 2hat"j" - hat"k"`
`vec"AB" xx vec"AC" = |(hat"i", hat"j", hat"k"),(-2, 0, -5),(1, -2, -1)|`
= `hat"i"(0 - 10) - hat"j"(2 + 5) + hat"k"(4 - 0)`
= `-10hat"i" - 7hat"j" + 4hat"k"`
`|vec"AB" xx vec"AC"| = |-10hat"i" - 7hat"j" + 4hat"k"|`
= `sqrt((-10)^2 + (-7)^2 + 4^2`
= `sqrt(100 + 49 + 16)`
= `sqrt(165)`
Area of the triangle ABC = `1/2 |vec"AB" xx vec"AC"|`
= `1/2 xx sqrt(165)`
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