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प्रश्न
Find the area of the triangle whose vertices are A(3, –1, 2), B(1, –1, –3) and C(4, –3, 1)
Using vectors, find the area of the triangle whose vertices are: A(3, –1, 2), B(1, –1, –3) and C(4, –3, 1)
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उत्तर
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"`
Area of ΔABC = `1/2 |vec"AB" xx vec"AC"|`
= `1/2 |vec"BA" xx vec"BC"|`
= `1/2 |vec"CA" xx vec"CB"|`
`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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