VectorsTS: Explicitly assessing information and drawing conclusions.
Warm Up:a) What are the coordinates of A and B b) What is the distance between A and B?
Vectors are directed line segments that are typically used to represent a force or speed (things that have both a magnitude and direction). The magnitude is indicated by the
length of the vector, and the direction is indicated by the slope
(or angle) of vector.
If a plane is flying due North East at 100mph, then it’s motion can be represented as a vector. Which is shown as
a terminating ray (or “directed segment”)
N100mph
45°
If a person is pushing a box up a hill which has an angle of elevation of 15° using 16 Newtons (N) of force to do it, then
it’s force can be shown using a vector, shown as this terminating ray (or “directed segment”)
16N
15°
Equivalent Directed Line Segments
BA
C
Component form of a Vector
The component form of a vector with initial point P (p1, p2) and terminal point Q (q1, q2) is given by
The magnitude of v is given by
SPECIAL VECTORS:
If ||v|| = 1, v is a unit vector.
If ||v|| = 0, v is the zero vector.
1 1 2 2 1 2, ,PQ q p q p v v v��������������
2 2 2 21 1 2 2 1 2( ) ( )q p q p v v v
v1
v2
||v||
P
Q
Ex: Find the component form and magnitude of the vector shown, then drawn an equal
vector whose terminal point is (5, 2)
Vector Operations: Scalar Multiplication
Given v find:
1) 2v
2) -v
3) 0.5v
1 2 1 2, ,k k u u ku ku u
Vector Operations: Addition
Given u and v
Find u + v
1 1 2 2,u v u v u v
u
v
Vector Operations: Subtraction
Find u - v
u
v
Example: Multiple Operations
Find u – 2v
u
v
Properties of Vector Addition and Scalar Multiplication
Let u, v, and w be vectors and let c and d be scalars. Then the following properties are true.
1) u + v = v + u
2) u + 0 = u
3) c(du) = (cd)u
4) c(u + v) = cu + cv
5) ||cv|| = |c| ||v||
6) (u + v) + w = u + (v + w)
7) u + (-u) = 0
8) (c + d)u = cu + du
9) 1(u) = u
10) 0(u) = 0
Unit Vectors
To find a unit vector divide the vector, v, by its magnitude.
This will have the same direction as the vector, v, but it’s magnitude (or length) will be 1.
unit vector v
v
Example:
Find a unit vector going in the same direction as < -2, 5 >. Verify it is a unit vector.
Writing a vector as a Linear Combination of Unit Vectors, &
Unit Vectors: = <1, 0> and = <0, 1>
Linear Combination of Unit Vectors
v = < v1, v2> = v1 + v2
i j
i j
i j
Example:
If u is a vector with initial side (2, -5) and terminal side (-1, 3), write u as a linear combination of the standard unit vectors and i j
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