The dot product of two vectors x → = [ x 1 x 2 ⋮ x n ] and y → = [ y 1 y 2 ⋮ y n ] in ℝ ″ is defined by x → ⋅ y → = x 1 y 1 + x 2 y 2 + ⋯ + x n y n .Note that the dot product of two vectors is a scalar.We say that the vectors x → and y → are perpendicular if x → ⋅ y → = 0 . Find all vectors in ℝ 3 perpendicular to [ 1 3 − 1 ] . Draw a sketch.
The dot product of two vectors x → = [ x 1 x 2 ⋮ x n ] and y → = [ y 1 y 2 ⋮ y n ] in ℝ ″ is defined by x → ⋅ y → = x 1 y 1 + x 2 y 2 + ⋯ + x n y n .Note that the dot product of two vectors is a scalar.We say that the vectors x → and y → are perpendicular if x → ⋅ y → = 0 . Find all vectors in ℝ 3 perpendicular to [ 1 3 − 1 ] . Draw a sketch.
Solution Summary: The author explains that the subspace spanned by R3 is perpendicular to the vector
The dot product of two vectors
x
→
=
[
x
1
x
2
⋮
x
n
]
and
y
→
=
[
y
1
y
2
⋮
y
n
]
in
ℝ
″
is defined by
x
→
⋅
y
→
=
x
1
y
1
+
x
2
y
2
+
⋯
+
x
n
y
n
.Note that the dot product of two vectors is a scalar.We say that the vectors
x
→
and
y
→
are perpendicular if
x
→
⋅
y
→
=
0
.
Find all vectors in
ℝ
3
perpendicular to
[
1
3
−
1
]
.
Draw a sketch.
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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