Mathematical Investigations IV
Name:
Vectors
Getting To the Point
Vector Concepts
A.
Vector notation and graphing:
1.
v = <a, b> corresponds to a directed segment that starts at the origin and ends at
the point (a, b).
2.
3.
4.
B.
a 
v = v = <a, b, c> = aiˆ  b ĵ  ck̂ = b , where a, b, and c are

c 

the x, y, and z-components of v, respectfully.
Vectors may be denoted by their components or by their direction and magnitude.
Attributes:
1.
v = <a, b> = < v cos , v sin  >, where  is the angle from the positive x-axis
to the vector.
2.
C.
Two vectors are equal if they have the same magnitude and the same direction.
Length or Magnitude: v  a 2 + b2
Vector Arithmetic:
1.
Addition: v + w = <v1+w1, v2+w2, v3+w3>. Vector addition can also be
accomplished graphically using triangulation or the parallelogram diagonal.
2.
Scalar multiplication: cv = <cv1, cv2, cv3>
3.
Subtraction: v – w = v + (-1)w.
4.
Dot product (yields a scalar): vw = = v w cos  = v1w1 + v2w2 + v3w3
iˆ
ĵ
k̂
Cross product (yields a vector): v  w = v1 v2 v3 = –w  v
5.
w1 w2
6.
w3
“Right-hand Rule” for Cross product
Vectors. 11.1
Rev. F08
Mathematical Investigations IV
Name:
D.
E.
Unit Vector:
1.
Magnitude = 1 unit
1 r
2.
v̂ = r v
v
3.
The basic unit vectors in 3-D space are
iˆ = <1, 0, 0>,
ĵ = <0, 1, 0>,


 wr cos v̂ = vv vwv
Miscellaneous
v w
v w
sin() =
v w
1.
cos() =
2.
Area of a triangle determined by v and w is
3.
Area of a parallelogram determined by v and w is v  w .
4.
Review matrix transformations.
5.
Vectors can be used to represent
a.
Force, such as weight and direction
b.
Velocity, such as speed and direction.
6.
Vector forces determine an equilibrium when the sum of the vectors is zero.
7.
v  w is orthogonal (i.e., perpendicular) to both v and w.
8.
v  w if and only if v  w = 0.
9.
v // w if and only if there exists some scalar k ≠ 0, such that k v = w.
v w
Vectors. 11.2

k̂ = <0, 0, 1>
Projections:
1. The “shadow” of one vector onto another.
2. projvw =
F.
and
1
 v w .
2
Rev. F08