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Mathematics
Mathematics
Problem
If
u
\mathbf{u}
u
,
v
\mathbf{v}
v
and
w
\mathbf{w}
w
is a linearly independent system of vectors, examine the system
p
\mathbf{p}
p
,
q
\mathbf{q}
q
and
r
\mathbf{r}
r
, where
p
=
(
cos
a
)
u
+
(
cos
b
)
v
+
(
cos
c
)
w
\mathbf{p} = (\cos a)\mathbf{u} + (\cos b)\mathbf{v} + (\cos c)\mathbf{w}
p
=
(
cos
a
)
u
+
(
cos
b
)
v
+
(
cos
c
)
w
,
q
=
(
sin
a
)
u
+
(
sin
b
)
v
+
(
sin
c
)
w
\mathbf{q} = (\sin a)\mathbf{u} + (\sin b)\mathbf{v} + (\sin c)\mathbf{w}
q
=
(
sin
a
)
u
+
(
sin
b
)
v
+
(
sin
c
)
w
,
r
=
sin
(
x
+
a
)
u
+
sin
(
x
+
b
)
v
+
sin
(
x
+
c
)
w
\mathbf{r} = \sin(x+a)\mathbf{u} + \sin(x+b)\mathbf{v} + \sin(x+c)\mathbf{w}
r
=
sin
(
x
+
a
)
u
+
sin
(
x
+
b
)
v
+
sin
(
x
+
c
)
w
for linearly dependent.
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