1. Example-1
1. Find VECTORTRIPLEPRODUCT(A,B,C)
`A=(1,2,3)`,`B=(4,5,6)`,`C=(2,6,5)`
Solution:
Here `vec A=(1,2,3),vec B=(4,5,6),vec C=(2,6,5)`
1. Calculate `Axx(BxxC)`
`vec B xx vec C`
`=|[i,j,k],[B_1,B_2,B_3],[C_1,C_2,C_3]|`
`=|[i,j,k],[4,5,6],[2,6,5]|`
`=i(5xx5-6xx6)-j(4xx5-6xx2)+k(4xx6-5xx2)`
`=i(25-36)-j(20-12)+k(24-10)`
`=-11i-8j+14k`
`=(-11,-8,14)`
`vec A xx vec (B xx C)`
`=|[i,j,k],[A_1,A_2,A_3],[(B xx C)_1,(B xx C)_2,(B xx C)_3]|`
`=|[i,j,k],[1,2,3],[-11,-8,14]|`
`=i(2xx14-3xx(-8))-j(1xx14-3xx(-11))+k(1xx(-8)-2xx(-11))`
`=i(28+24)-j(14+33)+k(-8+22)`
`=52i-47j+14k`
`=(52,-47,14)`
2. Calculate `(AxxB)xxC)`
`vec A xx vec B`
`=|[i,j,k],[A_1,A_2,A_3],[B_1,B_2,B_3]|`
`=|[i,j,k],[1,2,3],[4,5,6]|`
`=i(2xx6-3xx5)-j(1xx6-3xx4)+k(1xx5-2xx4)`
`=i(12-15)-j(6-12)+k(5-8)`
`=-3i+6j-3k`
`=(-3,6,-3)`
`vec (A xx B) xx vec C`
`=|[i,j,k],[(A xx B)_1,(A xx B)_2,(A xx B)_3],[C_1,C_2,C_3]|`
`=|[i,j,k],[-3,6,-3],[2,6,5]|`
`=i(6xx5-(-3)xx6)-j((-3)xx5-(-3)xx2)+k((-3)xx6-6xx2)`
`=i(30+18)-j(-15+6)+k(-18-12)`
`=48i+9j-30k`
`=(48,9,-30)`
2. Find VECTORTRIPLEPRODUCT(A,B,C)
`A=(5,-1,1)`,`B=(-2,0,4)`,`C=(3,4,5)`
Solution:
Here `vec A=(5,-1,1),vec B=(-2,0,4),vec C=(3,4,5)`
1. Calculate `Axx(BxxC)`
`vec B xx vec C`
`=|[i,j,k],[B_1,B_2,B_3],[C_1,C_2,C_3]|`
`=|[i,j,k],[-2,0,4],[3,4,5]|`
`=i(0xx5-4xx4)-j((-2)xx5-4xx3)+k((-2)xx4-0xx3)`
`=i(0-16)-j(-10-12)+k(-8-0)`
`=-16i+22j-8k`
`=(-16,22,-8)`
`vec A xx vec (B xx C)`
`=|[i,j,k],[A_1,A_2,A_3],[(B xx C)_1,(B xx C)_2,(B xx C)_3]|`
`=|[i,j,k],[5,-1,1],[-16,22,-8]|`
`=i((-1)xx(-8)-1xx22)-j(5xx(-8)-1xx(-16))+k(5xx22-(-1)xx(-16))`
`=i(8-22)-j(-40+16)+k(110-16)`
`=-14i+24j+94k`
`=(-14,24,94)`
2. Calculate `(AxxB)xxC)`
`vec A xx vec B`
`=|[i,j,k],[A_1,A_2,A_3],[B_1,B_2,B_3]|`
`=|[i,j,k],[5,-1,1],[-2,0,4]|`
`=i((-1)xx4-1xx0)-j(5xx4-1xx(-2))+k(5xx0-(-1)xx(-2))`
`=i(-4-0)-j(20+2)+k(0-2)`
`=-4i-22j-2k`
`=(-4,-22,-2)`
`vec (A xx B) xx vec C`
`=|[i,j,k],[(A xx B)_1,(A xx B)_2,(A xx B)_3],[C_1,C_2,C_3]|`
`=|[i,j,k],[-4,-22,-2],[3,4,5]|`
`=i((-22)xx5-(-2)xx4)-j((-4)xx5-(-2)xx3)+k((-4)xx4-(-22)xx3)`
`=i(-110+8)-j(-20+6)+k(-16+66)`
`=-102i+14j+50k`
`=(-102,14,50)`
This material is intended as a summary. Use your textbook for detail explanation.
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