Transcript Chapter 2

Chapter 2
Section 2.3
The Dot Product and the Cross
Product
Dot Product
The dot product of two vectors u and v is defined to be:
u ∙ v = u𝑇 v
This is true in every dimension in the case of 3 dimensions:
𝑢1
𝑣1
𝑢2 ∙ 𝑣2 = 𝑢1 𝑣1 + 𝑢2 𝑣2 + 𝑢3 𝑣3
𝑢3
𝑣3
Geometrically the dot product gives information about the
angle  between the vectors u and v, specifically:
u ∙ v = u v cos 𝜃
u

v
Orthogonal Vectors
If u and v are nonzero vectors the only way that u v cos 𝜃 = 0 is to have cos 𝜃 = 0
which means that 𝜃 = 𝜋2 (or 90°). This means the vectors are perpendicular which we call
orthogonal. The vector u is orthogonal to vector v if and only if u ∙ v = 0.
Algebraic Properties of Dot Product
Let u, v, and w be vectors and c a scalar.
a) u ∙ u = u 2 ≥ 0 and u ∙ u = 0 if and only if u = 𝜽 (Here  is the zero vector.)
b) If u and v are nonzero vectors u ∙ v = 0 if and only if u ⊥ v
c) u ∙ v = v ∙ u
d) 𝑐u ∙ v = u ∙ 𝑐v = 𝑐 u ∙ v
e) u ∙ v + w = u ∙ v + u ∙ w
Example
Find the angle  between the
vectors u and v given as:
−4
3
u = 0 and v = 6
3
−2
u = 16 + 0 + 9 = 5
v = 9 + 36 + 4 = 7
u ∙ v = −12 + 0 − 6 = −18
−18 = 5 ∙ 7 cos 𝜃
−18
cos 𝜃 =
35
−1 −18
𝜃 = cos 35
≈ 2.11097 ≈ 121°
Projections
The projection of a vector u onto a non zero vector v is a
vector parallel to v whose difference with u is orthogonal to v.
u

𝑝𝑟𝑜𝑗v u
v
To derive a formula for this let h be the length that v must be
rescaled to get an orthogonal vector.
h
ℎ
u∙v
= cos 𝜃 or ℎ = u cos 𝜃
𝑝𝑟𝑜𝑗v u =
v
u
v∙v
v
v
=
v = A unit vector in the direction of v
Example
v
v∙v
9
2
u = 4 and v = −1
Multiply the unit vector in v’s direction by h to get the
6
1
20
projection.
3
2
ℎ v
u v cos 𝜃
u∙v
20
10
𝑝𝑟𝑜𝑗v u = 6 −1 = − 3
𝑝𝑟𝑜𝑗v u =
v=
v=
v
v∙v
v∙v
v∙v
10
1
3
Determinants of 2 × 2 Matrices
The determinant of a square matrix M is a scalar (number)
that is denoted by 𝑀 and for 2 × 2 matrices can be
computed by the formula to the right.
𝑎
𝑐
For a 𝑛 × 𝑛 matrix M the 𝑛 − 1 × 𝑛 − 1 minor matrix 𝑀𝑖𝑗 comes
from deleting the ith row and jth column of the matrix M. For the matrix
M given to the right we compute minor matrices 𝑀11 , 𝑀12 , 𝑀13 .
𝑀11
𝑎
= 𝑑
𝑔
𝑏
𝑒
ℎ
𝑐
𝑓 = 𝑒
ℎ
𝑖
𝑎
𝑓
, 𝑀12 = 𝑑
𝑖
𝑔
𝑏
𝑒
ℎ
𝑐
𝑑
𝑓 =
𝑔
𝑖
𝑎
𝑓
, 𝑀13 = 𝑑
𝑖
𝑔
𝑏
= 𝑎𝑑 − 𝑏𝑐
𝑑
𝑎
𝑀= 𝑑
𝑔
𝑏
𝑒
ℎ
𝑏
𝑒
ℎ
𝑐
𝑓
𝑖
𝑐
𝑑
𝑓 =
𝑔
𝑖
𝑒
ℎ
Determinants of 3 × 3 Matrices
The determinant of a 3 × 3 matrix can be found by expanding by cofactors, signed
determinants of minor matrices multiplied by the entry in the row or column, down any
row or column. For convenience we usually express the formula in term of an expansion in
the first row.
𝑎 𝑏 𝑐
𝑑 𝑓
𝑑 𝑒
𝑑 𝑒 𝑓 = 𝑀11 𝑎 − 𝑀12 𝑏 + 𝑀13 𝑐 = 𝑒 𝑓 𝑎 −
𝑏+
𝑐
𝑔 ℎ
𝑔 𝑖
ℎ
𝑖
𝑔 ℎ 𝑖
Cross Products of Vectors
The cross product of two vectors u and v is another vector and is denoted u × v. It can be
computed forming a matrix with first row i j k second row u𝑇 and third row v 𝑇 and
computing the determinant by expanding across the first row. This formula is given below.
𝑢1
𝑣1
i
𝑢2 × 𝑣2 = 𝑢1
𝑢3
𝑣3
𝑣1
j
𝑢2
𝑣2
3
2
Example: Find 1 × 0
−1
1
k
𝑢
𝑢3 = 𝑣2
2
𝑣3
𝑢3
𝑢1
i
−
𝑣3
𝑣1
i
3
2
1 × 0 = 3
−1
1
2
𝑢3
𝑢1
j
+
𝑣3
𝑣1
𝑢2 𝑣3 − 𝑢3 𝑣2
𝑢2
− 𝑢1 𝑣3 − 𝑢3 𝑣1
𝑣2 k =
𝑢1 𝑣2 − 𝑢2 𝑣1
j
k
1−0
1 −1 = − 3 − −2
0−2
0 1
1
= −5
−2
Notice what happens if we take the result and dot it with each vector we started with.
3
1
2
1
1 ∙ −5 = 3 − 5 + 2 = 0 and 0 ∙ −5 = 2 + 0 − 2 = 0
−1
−2
1
−2
This means that both of the vectors that we started with are orthogonal (perpendicular)
to the result. This is not a coincidence. If you worked through the algebra you would find
that no matter what two vectors u and v you start with the result of the cross product u ×
v is a vector perpendicular to u and v. This gives us an idea of geometrically how u × v is
related to u and v.
Geometric Properties of Cross Product
Direction:
We know that the vector u × v is orthogonal to both u
and v but, there are 2 directions that are perpendicular to
both vectors. It is the vector in the direction of the right
hand rule. That is your right hand placed at the vertex so
your fingers curl from u to v.
Length:
The length of the cross product of u and v use a great
deal of algebra to derive but we get the following
relation:
u × v = u v sin 𝜃
Where  is the angle between the vectors u and v. This
can be thought of more geometrically by letting h be the
height of the parallelogram formed with sides u and v.
u×v
v

h
u
ℎ
= sin 𝜃 or ℎ = v sin 𝜃
v
ℎ u = u v sin 𝜃
ℎ u = u×v
ℎ u = Area of parallelogram
u × v =Area of parallelogram
Algebraic Properties of Cross Product (Let u, v and w be vectors and c a scalar)
a. u × v = − v × u
b. u × v = 0 if and only if u is parallel to v
c. 𝑐u × v = u × 𝑐v = 𝑐 u × v
d. u × v + w = u × v + u × w
e. u ∙ u × v = 0 = v ∙ u × v