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Dot product and vector projections (Sect. 12.3)
I (^) Two definitions for the dot product.
I (^) Geometric definition of dot product.
I (^) Orthogonal vectors.
I (^) Dot product and orthogonal projections.
I (^) Properties of the dot product.
I (^) Dot product in vector components.
I (^) Scalar and vector projection formulas.
Two main ways to introduce the dot product
Geometrical
definition
→ Properties →
Expression in
components.
Definition in
components
→ Properties →
Geometrical
expression.
We choose the first way, the textbook chooses the second way.
Dot product and vector projections (Sect. 12.3)
I (^) Two definitions for the dot product.
I (^) Geometric definition of dot product.
I (^) Orthogonal vectors.
I (^) Dot product and orthogonal projections.
I (^) Properties of the dot product.
I (^) Dot product in vector components.
I (^) Scalar and vector projection formulas.
The dot product of two vectors is a scalar
Definition
The dot product of the vectors v and w in Rn, with n = 2, 3, having magnitudes |v |, |w| and angle in between θ, where 0 ≤ θ ≤ π, is denoted by v · w and given by
v · w = |v | |w| cos(θ).
O
V
W
Initial points together.
Perpendicular vectors have zero dot product.
Definition
Two vectors are perpendicular, also called orthogonal, iff the angle in between is θ = π/2.
0 = / 2
V W
Theorem
The non-zero vectors v and w are perpendicular iff v · w = 0.
Proof.
0 = v · w = |v| |w| cos(θ)
|v| 6 = 0, |w| 6 = 0
cos(θ) = 0
0 6 θ 6 π
⇔ θ =
π
2
The dot product of i, j and k is simple to compute
Example
Compute all dot products involving the vectors i, j , and k.
Solution: Recall: i = 〈 1 , 0 , 0 〉, j = 〈 0 , 1 , 0 〉, k = 〈 0 , 0 , 1 〉.
x
i j
k
z
y
i · i = 1, j · j = 1, k · k = 1,
i · j = 0, j · i = 0, k · i = 0,
i · k = 0, j · k = 0, k · j = 0. C
Dot product and vector projections (Sect. 12.3)
I (^) Two definitions for the dot product.
I (^) Geometric definition of dot product.
I (^) Orthogonal vectors.
I (^) Dot product and orthogonal projections.
I (^) Properties of the dot product.
I (^) Dot product in vector components.
I (^) Scalar and vector projection formulas.
The dot product and orthogonal projections.
Remark: The dot product is closely related to orthogonal
projections of one vector onto the other.
Recall: v · w = |v| |w| cos(θ).
O W
V
|v| cos(θ) =
v · w
|w|
O W
V
|w| cos(θ) =
v · w
|v|
Remark: If |u| = 1, then v · u is the projection of v along u.
Properties of the dot product.
(c) u · (v + w) = u · v + u · w, is non-trivial. The proof is:
W
w
V+W
|V+W| cos( 0 )
V
0 (^0) W
|V| cos( 0 )V
|W| cos( 0 )W
U
(^0) V
|v + w| cos(θ) =
u · (v + w) |u|
,
|w| cos(θw ) =
u · w |u|
,
|v| cos(θv ) =
u · v |u|
,
⇒ u · (v + w) = u · v + u · w
Dot product and vector projections (Sect. 12.3)
I (^) Two definitions for the dot product.
I (^) Geometric definition of dot product.
I (^) Orthogonal vectors.
I (^) Dot product and orthogonal projections.
I (^) Properties of the dot product.
I (^) Dot product in vector components.
I (^) Scalar and vector projection formulas.
The dot product in vector components (Case R
2
Theorem
If v = 〈vx , vy 〉 and w = 〈wx , wy 〉, then v · w is given by
v · w = vx wx + vy wy.
Proof.
Recall: v = vx i + vy j and w = wx i + wy j. The linear property of the dot product implies
v · w = (vx i + vy j ) · (wx i + wy j )
v · w = vx wx i · i + vx wy i · j + vy wx j · i + vy wy j · j.
Recall: i · i = j · j = 1 and i · j = j · i = 0. We conclude that
v · w = vx wx + vy wy.
The dot product in vector components (Case R
3
Theorem
If v = 〈vx , vy , vz 〉 and w = 〈wx , wy , wz 〉, then v · w is given by
v · w = vx wx + vy wy + vz wz.
I (^) The proof is similar to the case in R^2.
I (^) The dot product is simple to compute from the vector
component formula v · w = vx wx + vy wy + vz wz.
I (^) The geometrical meaning of the dot product is simple to see
from the formula v · w = |v| |w| cos(θ).
Scalar and vector projection formulas.
Theorem
The scalar projection of v along w is the number pw (v ),
pw (v ) =
v · w
|w|
The vector projection of v along w is the vector pw (v ),
pw (v ) =
v · w
|w|
w
|w|
| W |
O
| W |
w
V
W
P ( V ) = ( V W ) W | W |
O
| W |
w
V
W
P ( V ) = ( V W ) W
Example
Find the scalar projection of b = 〈− 4 , 1 〉 onto a = 〈 1 , 2 〉.
Solution: The scalar projection of b onto a is the number
pa(b) = |b| cos(θ) =
b · a
|a|
12 + 2^2
We therefore obtain pa(b) = −
x
p (b)
a
b
a
y
Example
Find the vector projection of b = 〈− 4 , 1 〉 onto a = 〈 1 , 2 〉.
Solution: The vector projection of b onto a is the vector
pa(b) =
b · a
|a|
a
|a|
We therefore obtain pa(b) = −
x
p (b)
a
b
a
y
Example
Find the vector projection of a = 〈 1 , 2 〉 onto b = 〈− 4 , 1 〉.
Solution: The vector projection of a onto b is the vector
pb(a) =
a · b
|b|
b
|b|
We therefore obtain pa(b) =
x
b
a
p (a)
b
y
