1. Find Area of triangle determined by two vectors
`A=(1,2,3)`, `B=(4,5,6)`

Solution:
Here `vec A=(1,2,3),vec B=(4,5,6)`

Area `=1/2|vec A xx vec B|`

1. Calculate cross product
`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)`

2. Calculate magnitude
`|vec vec A xx vec B|`

`=sqrt((-3)^2 + 6^2 + (-3)^2)`

`=sqrt(9 + 36 + 9)`

`=sqrt(54)`

`=3sqrt(6)`


3. Calculate triangle area
Area `=1/2 * 3sqrt(6)=(3sqrt(6))/2`

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2. Find Area of triangle determined by two vectors
`A=(5,-1,1)`, `B=(-2,3,4)`

Solution:
Here `vec A=(5,-1,1),vec B=(-2,3,4)`

Area `=1/2|vec A xx vec B|`

1. Calculate cross product
`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],[-2,3,4]|`

`=i((-1)xx4-1xx3)-j(5xx4-1xx(-2))+k(5xx3-(-1)xx(-2))`

`=i(-4-3)-j(20+2)+k(15-2)`

`=-7i-22j+13k`

`=(-7,-22,13)`

2. Calculate magnitude
`|vec vec A xx vec B|`

`=sqrt((-7)^2 + (-22)^2 + 13^2)`

`=sqrt(49 + 484 + 169)`

`=sqrt(702)`

`=3sqrt(78)`


3. Calculate triangle area
Area `=1/2 * 3sqrt(78)=(3sqrt(78))/2`

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