1. Example-1
1. Find VolumePyramid(A,B,C)
`A=(1,2,3)`,`B=(3,0,6)`,`C=(7,1,9)`
Solution:
Here `vec A=(1,2,3),vec B=(3,0,6),vec C=(7,1,9)`
Volume `=1/3|vec A * (vec B xx vec C)|`
1. Calculate scalar triple product
`A*(B xx C)`
`=|[A_1,A_2,A_3],[B_1,B_2,B_3],[C_1,C_2,C_3]|`
`=|[1,2,3],[3,0,6],[7,1,9]|`
`=1(0xx9-6xx1)-2(3xx9-6xx7)+3(3xx1-0xx7)`
`=1(0-6)-2(27-42)+3(3-0)`
`=1(-6)-2(-15)+3(3)`
`=-6+30+9`
`=33`
2. Calculate pyramid volume
Volume `=1/3 * 33=11`
2. Find VolumePyramid(A,B,C)
`A=(5,-1,1)`,`B=(-2,3,4)`,`C=(3,4,5)`
Solution:
Here `vec A=(5,-1,1),vec B=(-2,3,4),vec C=(3,4,5)`
Volume `=1/3|vec A * (vec B xx vec C)|`
1. Calculate scalar triple product
`A*(B xx C)`
`=|[A_1,A_2,A_3],[B_1,B_2,B_3],[C_1,C_2,C_3]|`
`=|[5,-1,1],[-2,3,4],[3,4,5]|`
`=5(3xx5-4xx4)--1((-2)xx5-4xx3)+1((-2)xx4-3xx3)`
`=5(15-16)--1(-10-12)+1(-8-9)`
`=5(-1)--1(-22)+1(-17)`
`=-5-22-17`
`=-44`
2. Calculate pyramid volume
Volume `=1/3 * 44=44/3`
This material is intended as a summary. Use your textbook for detail explanation.
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