2. Example-2
1. Find VECTORTRIPLEPRODUCT(A,B,C)
`A=(1,2,3)`,`B=(-1,0,6)`,`C=(3,4,5)`
Solution:
Here `vec A=(1,2,3),vec B=(-1,0,6),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],[-1,0,6],[3,4,5]|`
`=i(0xx5-6xx4)-j((-1)xx5-6xx3)+k((-1)xx4-0xx3)`
`=i(0-24)-j(-5-18)+k(-4-0)`
`=-24i+23j-4k`
`=(-24,23,-4)`
`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],[-24,23,-4]|`
`=i(2xx(-4)-3xx23)-j(1xx(-4)-3xx(-24))+k(1xx23-2xx(-24))`
`=i(-8-69)-j(-4+72)+k(23+48)`
`=-77i-68j+71k`
`=(-77,-68,71)`
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],[-1,0,6]|`
`=i(2xx6-3xx0)-j(1xx6-3xx(-1))+k(1xx0-2xx(-1))`
`=i(12-0)-j(6+3)+k(0+2)`
`=12i-9j+2k`
`=(12,-9,2)`
`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],[12,-9,2],[3,4,5]|`
`=i((-9)xx5-2xx4)-j(12xx5-2xx3)+k(12xx4-(-9)xx3)`
`=i(-45-8)-j(60-6)+k(48+27)`
`=-53i-54j+75k`
`=(-53,-54,75)`
2. Find VECTORTRIPLEPRODUCT(A,B,C)
`A=(5,6,1)`,`B=(0,2,3)`,`C=(3,-1,5)`
Solution:
Here `vec A=(5,6,1),vec B=(0,2,3),vec C=(3,-1,5)`
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],[0,2,3],[3,-1,5]|`
`=i(2xx5-3xx(-1))-j(0xx5-3xx3)+k(0xx(-1)-2xx3)`
`=i(10+3)-j(0-9)+k(0-6)`
`=13i+9j-6k`
`=(13,9,-6)`
`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],[5,6,1],[13,9,-6]|`
`=i(6xx(-6)-1xx9)-j(5xx(-6)-1xx13)+k(5xx9-6xx13)`
`=i(-36-9)-j(-30-13)+k(45-78)`
`=-45i+43j-33k`
`=(-45,43,-33)`
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],[5,6,1],[0,2,3]|`
`=i(6xx3-1xx2)-j(5xx3-1xx0)+k(5xx2-6xx0)`
`=i(18-2)-j(15-0)+k(10-0)`
`=16i-15j+10k`
`=(16,-15,10)`
`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],[16,-15,10],[3,-1,5]|`
`=i((-15)xx5-10xx(-1))-j(16xx5-10xx3)+k(16xx(-1)-(-15)xx3)`
`=i(-75+10)-j(80-30)+k(-16+45)`
`=-65i-50j+29k`
`=(-65,-50,29)`
This material is intended as a summary. Use your textbook for detail explanation.
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