Expressions of the form u × ( v × w ) and ( u × v ) × w are called vector triple products. It can be proved with some effort that u × ( v × w ) = ( u ⋅ w ) v − ( u ⋅ v ) w ( u × v ) × w = ( w ⋅ u ) v − ( w ⋅ v ) u These expressions can be summarized with the following memonic rule: See if you can figure out what the expressions “outer,” “remote,” and “adjacent” mean in this rule, and then use the rule to find the two vector triple of the vectors u = i + 2 j − k , v = − 3 i + j + 2 k , w = 2 i − j + 4 k
Expressions of the form u × ( v × w ) and ( u × v ) × w are called vector triple products. It can be proved with some effort that u × ( v × w ) = ( u ⋅ w ) v − ( u ⋅ v ) w ( u × v ) × w = ( w ⋅ u ) v − ( w ⋅ v ) u These expressions can be summarized with the following memonic rule: See if you can figure out what the expressions “outer,” “remote,” and “adjacent” mean in this rule, and then use the rule to find the two vector triple of the vectors u = i + 2 j − k , v = − 3 i + j + 2 k , w = 2 i − j + 4 k
Solution Summary: The author explains the formulae used to calculate the triple vector products for the vectors u,
u
×
(
v
×
w
)
and
(
u
×
v
)
×
w
are called vector triple products. It can be proved with some effort that
u
×
(
v
×
w
)
=
(
u
⋅
w
)
v
−
(
u
⋅
v
)
w
(
u
×
v
)
×
w
=
(
w
⋅
u
)
v
−
(
w
⋅
v
)
u
These expressions can be summarized with the following memonic rule:
See if you can figure out what the expressions “outer,” “remote,” and “adjacent” mean in this rule, and then use the rule to find the two vector triple of the vectors
u
=
i
+
2
j
−
k
,
v
=
−
3
i
+
j
+
2
k
,
w
=
2
i
−
j
+
4
k
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Finite Mathematics and Calculus with Applications (10th Edition)
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