1. Digital Lesson
Vectors

2. A vector is a quantity with both a magnitude and a direction.
A ball flies through the air at a certain speed and in a particular direction. The speed and direction are the velocity of the ball. The velocity is a vector quantity since it has both a magnitude and a direction.
Vectors are used to represent velocity, force, tension, and many other quantities.
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Definition: Vector

3. Two vectors, u and v, are equal if the line segments
A quantity with magnitude and direction is represented by a directed line segment PQ. P Q
Initial point
Terminal point
The vector v = PQ is the set of all directed line segments of length ||PQ|| which are parallel to PQ.
Two vectors, u and v, are equal if the line segments
representing them are parallel and have the same length or magnitude. v u
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Directed Line Segment

4. The component form of vector v is written
A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (v1, v2). x y P Q
(v1, v2)
Components of v
The component form of vector v is written
If point P is the origin and point Q = (5, 6), then v =
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Standard Position

5. Definition: Component Form and Magnitude
If v is a vector with initial point P = (p1 , p2) and terminal point Q = (q1 , q2), then
1. The component form of v is
2. The magnitude (or length) of v is x y P Q
(p1, p2)
(q1, q2)
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Definition: Component Form and Magnitude

6. Example: Component Form and Magnitude
Find the component form and magnitude of vector v that has initial point (2, 2) and terminal point (–1, 4). x y -2 2
The component form of v is
3.61 units
Q = (–1, 4)
P = (2, 2)
The magnitude of v is
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Example: Component Form and Magnitude

7. Vector Addition and Subtraction
Example: Given vectors
find u + v and u – v .
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Vector Addition and Subtraction

8. Scalar Multiplication
Scalar multiplication is the product of a scalar, or real number, times a vector.
Example: Given vector
find –2u. x -4 4 -8 y u
–2u
The product of –2 and u gives a vector twice as long as and in the
opposite direction of u.
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Scalar Multiplication

9. Find a unit vector in the direction of v = –1, 1.
Unit Vectors
A unit vector in the direction of v is a vector, u, that has a magnitude of 1 and the same direction as v.
Divide v by its length to obtain a unit vector.
Example:
Find a unit vector in the direction of v = –1, 1.
unit vector in the direction of v
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Unit Vectors

10.  is the direction angle of the vector u.
Direction Angles
If u is a unit vector such that  is the angle (measured counterclockwise) from the positive x-axis to u, the terminal point of u lies on the unit circle and y x 1 –1
– 1
(x, y)
y = sin 
x = cos   u
 is the direction angle of the vector u.
Continued.
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Direction Angles

11. The direction angle  for v is determined by
Direction Angles continued
If v = ai + bj is any vector that makes an angle  with the positive x-axis, then it has the same direction as u and
The direction angle  for v is determined by
Example: The direction angle of u = 3i +3j is
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Direction Angles

12. Definition: Dot Product
This yields a scalar.
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Definition: Dot Product

13. Properties: Dot Product
Dot Product Properties:
Let u, v, and w be vectors and let c be a scalar.
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Properties: Dot Product

14. Example:
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Example: Dot Product

15. Definition: Angle Between Two Vectors
The angle between two nonzero vectors is the angle , between their standard position vectors.
Origin 
v - u v u
Find the angle between the vectors and
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Definition: Angle Between Two Vectors

16. Definition: Orthogonal Vectors
The vectors u and v are orthogonal if
Example: x
– 2 2 y 4
Because the dot product is 0, vectors u and v are orthogonal.
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Definition: Orthogonal Vectors