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what is the area of vertices of triangle (0,0,0),(2,3,5)&(5,6,9)

what is the area of vertices of triangle (0,0,0),(2,3,5)&(5,6,9)

Grade:upto college level

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
7 years ago
The area of a ? in 3-space can be found by determining ½ of the magnitude of the 'cross-product' of any two of the three vectors, V1, V2, V3 representing the sides of the 3-D ?

(I will use the symbol ? to denote a vector, if it is not clear from the context)

A(3D?) = ½ | V1 X V2 | ... OR ... = ½ | V1 X V3 | ... OR ... = ½ | V2 X V3 |,
where 'X' denotes 'cross product'

Any of the three vectors, say for example ?V1 = ?PQ, representing one of the sides of the 3-D ?, can be determined by the vector subtraction of the two Origin-Vertex vectors, ?OP – ?OQ, which can be easily determined by the subtraction of the corresponding pairs of co-ordinates.

V1 = ?PQ = ?OP – ?OQ = V1[-2, -3, –5]
V2 = ?PR = ?OP – ?OR = V2[–5, -6, –9]
A(?PQR) = ½ | ?PQ X ?PR | = ½ | V1 X V2 |
= ½ | [-2, -3, –5]X [–5, -6, –9] |
= ½ | [27-30, 18-25, 12-15] |
= ½ | [-3, -7, -3] |
= ½ √{3² +(7)² + 3²}
= ½(8.185)
= 4.0925 units2

Thanks and Regards
Shaik Aasif
askIITians Faculty

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