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Vector Algebra

Question
CBSEENMA12034228

Find the area of the triangle (by vectors) with vertices
A (3, – 1, 2), B (1, – 1, – 3) and C (4, – 3, 1).

Solution
Given vertices are A (3, – 1, 2) , B (1, – 1, – 3), C (4, – 3, 1 ).
Let straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top be position vectors of A, B, C respectively.
therefore space space space straight a with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space straight j with hat on top space plus space 2 space straight k with hat on top comma space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space minus space 3 space straight k with hat on top comma space space space straight c with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space straight k with hat on top
Now,       BC with rightwards arrow on top space equals space straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight B space equals space straight c with rightwards arrow on top space minus space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 4 space straight k with hat on top
              BA with rightwards arrow on top space equals space straight P. straight V. space of space straight A space minus space straight P. straight V. space of space straight B space equals space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space equals 2 space straight i with hat on top space plus space 5 space straight k with hat on top
therefore space space space BC with rightwards arrow on top cross times BA with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 3 cell negative 2 end cell 4 row 2 0 5 end table close vertical bar space equals space straight i with hat on top space open vertical bar table row cell negative 2 end cell 4 row cell space space space 0 end cell 5 end table close vertical bar space minus straight j with hat on top space open vertical bar table row 3 4 row 2 5 end table close vertical bar plus straight k with hat on top space open vertical bar table row 3 cell negative 2 end cell row 2 cell space space space 0 end cell end table close vertical bar
space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis negative 10 minus 0 right parenthesis space straight i with hat on top space minus space left parenthesis 15 minus 8 right parenthesis space straight j with hat on top space plus space left parenthesis 0 plus 4 right parenthesis space straight k with hat on top space equals space minus 10 space straight i with hat on top space minus space 7 space straight j with hat on top space plus space 4 space straight k with hat on top
open vertical bar BC with rightwards arrow on top space cross times space BA with rightwards arrow on top close vertical bar space equals space square root of 100 plus 49 plus 16 end root space equals space square root of 165
Area space of space increment ABC space equals space 1 half space open vertical bar BC with rightwards arrow on top space cross times space BA with rightwards arrow on top space space close vertical bar space equals space 1 half square root of 165 space sq. space units