Precalc 6.3 Notes.notebook
LESSON 6.3 Vectors in the Plane
Vectors are representations of quantities that have both magnitude and direction. A vector is denoted by a lowercase, boldface letter such as u or v.
Q
The directed line segment, or vector, has initial point P and terminal point Q.
The magnitude, or length, of can be found using the distance formula and is denoted by
P
Two vectors that have the same magnitude and direction are equivalent.
Precalc 6.3 Notes.notebook
EX. 1 Show that the two vectors, u and v are equivalent.
u
y
5
v
(4,4)
S
4
Magnitude
v
3
2
(1,2)
R
(3,2)
Q
u
1
Direction
x
(0,0)
P
0
1
2
3
4
5
Precalc 6.3 Notes.notebook
Component Form of a Vector
The component form of a vector with initial point P=(x1,y1) and terminal point Q=(x2,y2) is given by
v
The magnitude (or length) of v is given by
v
If v , v is a unit vector.
v if and only if v is the zero vector 0.
Precalc 6.3 Notes.notebook
EX. 2 Finding the Component Form of a Vector
Find the component form and magnitude of a vector v that has initial point (4,­7) and terminal point (­1,5).
Precalc 6.3 Notes.notebook
Vector Operations
Let u
and v
(a real number).
Scalar Multiplication
u
Vector Addition
u
v
Vector Subtraction
u
v
be vectors and let k be a scalar Graphically
Precalc 6.3 Notes.notebook
EX. 3 Let v vectors.
a.
Vector Operations
and w
, and find each of the following b.
Precalc 6.3 Notes.notebook
Properties of Vector Addition and Scalar Multiplication
Let u, v, and w be vectors and c and d be scalars.
Precalc 6.3 Notes.notebook
Unit Vectors
The unit vector in the direction of v is given by
u
Note that u is a scalar multiple of v.
The vector u has a magnitude of 1 and is in the same direction as v.
Precalc 6.3 Notes.notebook
EX. 4 Finding the Unit Vector
Find a unit vector in the direction of v
result has a magnitude of 1.
and verify that the Precalc 6.3 Notes.notebook
Standard Unit Vectors
and
The unit vectors
vectors and are denoted by
i
are called the standard unit j
These vectors can be used to represent any vector v
as follows
v
This representation of the vector v is called a linear combination of the vectors i and j.
The scalar v1 is the horizontal component of v, and the scalar v2 is the vertical component of v.
Precalc 6.3 Notes.notebook
EX. 5 Writing a Linear Combination of Unit Vectors
Let u be the vector with initial point (2,­5) and terminal point (­1,3). Write u as a linear combination of the standard unit vectors i and j.
y
6
5
4
3
2
1
­6
­5
­4
­3
­2
­1
0
­1
­2
­3
­4
­5
­6
x
1
2
3
4
5
6
Precalc 6.3 Notes.notebook
Direction Angles
v
EX. 6 Finding Direction Angles
y
5
Find the direction angle of v=3i+3j.
4
3
v
2
1
x
­5
­4
­3
­2
­1
0
­1
­2
­3
­4
­5
1
2
3
4
5