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Mathematics
Mathematics
Problem
57. Properties of the Dot Product Let
u
,
v
\mathbf{u}, \mathbf{v}
u
,
v
, and
w
\mathbf{w}
w
be vectors and let
c
c
c
be a scalar. Use the component form of vectors to prove the following properties. (a)
0
⋅
u
=
0
\mathbf{0} \cdot \mathbf{u}=0
0
⋅
u
=
0
(b)
u
⋅
(
v
+
w
)
=
u
⋅
v
+
u
⋅
w
\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}
u
⋅
(
v
+
w
)
=
u
⋅
v
+
u
⋅
w
(c)
(
u
+
v
)
⋅
w
=
u
⋅
w
+
v
⋅
w
(\mathbf{u}+\mathbf{v}) \cdot \mathbf{w}=\mathbf{u} \cdot \mathbf{w}+\mathbf{v} \cdot \mathbf{w}
(
u
+
v
)
⋅
w
=
u
⋅
w
+
v
⋅
w
(d)
(
c
u
)
⋅
v
=
u
⋅
(
c
v
)
=
c
(
u
⋅
v
)
(c \mathbf{u}) \cdot \mathbf{v}=\mathbf{u} \cdot(c \mathbf{v})=c(\mathbf{u} \cdot \mathbf{v})
(
c
u
)
⋅
v
=
u
⋅
(
c
v
)
=
c
(
u
⋅
v
)
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