Let θ be a fixed real number and let x 1 = ( cos θ sin θ ) and x 2 = ( − sin θ cos θ ) Show that { x 1 , x 2 } is an orthonormal basis for ℝ 2 . Given a vector y in ℝ 2 , write it as a linear combination c 1 x 1 + c 2 x 2 . Verify that c 1 2 + c 2 2 = ‖ y ‖ 2 = y 1 2 + y 2 2
Let θ be a fixed real number and let x 1 = ( cos θ sin θ ) and x 2 = ( − sin θ cos θ ) Show that { x 1 , x 2 } is an orthonormal basis for ℝ 2 . Given a vector y in ℝ 2 , write it as a linear combination c 1 x 1 + c 2 x 2 . Verify that c 1 2 + c 2 2 = ‖ y ‖ 2 = y 1 2 + y 2 2
Solution Summary: The author explains how to calculate leftx_1, if their dot product equals zero.
Let
θ
be a fixed real number and let
x
1
=
(
cos
θ
sin
θ
)
and
x
2
=
(
−
sin
θ
cos
θ
)
Show that
{
x
1
,
x
2
}
is an orthonormal basis for
ℝ
2
.
Given a vector y in
ℝ
2
,
write it as a linear combination
c
1
x
1
+
c
2
x
2
.
Verify that
c
1
2
+
c
2
2
=
‖
y
‖
2
=
y
1
2
+
y
2
2
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.
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