Vector Algebra And Calculus Cheat Sheet Page 7
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2
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28Vector triple product a × (b × c)
2.6
a
b
x
c
In arbitrary direction
a × (b × c) is perpendicular to (b × c)
c
but (b × c) is perpendicular to b and c.
So a × (b × c) must be coplanar with b and c.
b
⇒a × (b × c) = λb + µc
a
x
( b
x c )
(a × (b × c))
= a
(b × c)
a
(b × c)
1
2
3
3
2
= a
(b
c
b
c
) + a
(b
c
b
c
)
2
1
2
2
1
3
1
3
3
1
= (a
c
+ a
c
)b
(a
b
+ a
b
)c
2
2
3
3
1
2
2
3
3
1
= (a
c
+ a
c
+ a
c
)b
(a
b
+ a
b
+ a
b
)c
1
1
2
2
3
3
1
1
1
2
2
3
3
1
= (a · c)b
(a · b)c
1
1
Similarly for components 2 and 3: so
[a × (b × c)] = (a · c)b
(a · b)c
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