1. Vectors in the Plane

2. Magnitude and Direction
A vector is a quantity with both a magnitude and a direction.
Magnitude and 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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3. Magnitude of a Vector:
Direction of a Vector:

4. 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 with initial point P and terminal point Q. P Q
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

5. Scalar Multiplication
Scalar multiplication is the product of a scalar, or real number, times a vector.
Scalar Multiplication
For example, the scalar 3 times v results in the vector 3v, three times as long and in the same direction as v. v 3v
The product of and v gives a vector half as long as and in the opposite direction to v. v
- v
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6. 1. Place the initial point of v at the terminal point of u.
Vector Addition
Vector Addition v u
To add vectors u and v:
1. Place the initial point of v at the terminal point of u.
2. Draw the vector with the same initial point as u and the same terminal point as v. v u u v
u + v
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7. To subtract vectors u and v:
Vector Subtraction
Vector Subtraction v u
To subtract vectors u and v:
1. Place the initial point of v at the initial point of u.
2. Draw the vector u  v from the terminal point of v to the terminal point of u. v u v u
u  v
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8. The new vector is called the resultant or displacement vector.
Vector Operations
To add vectors in component form, just add the horizontal components and the vertical components.
To add vectors graphically, just play “follow the leader.” Then draw a new vector from the start of the first to the end of the second.
The new vector is called the resultant or displacement vector.

9. To multiply a vector by a scalar (constant), just distribute the number to both coordinates.
Graphically, you make the vector that many times as long in the same direction
To make a negative vector (for subtraction) just distribute a negative.
Graphically, you have the same slope and magnitude. You just go in the opposite direction.

10. A Bigger Example 2u
-3v

11. 1. The component form of v is v = q1  p1, q2  p2
A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (u1, u2).
Standard Position x y
(u1, u2)
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
v = q1  p1, q2  p2
2. The magnitude (or length) of v is
||v|| = x y
P (p1, p2)
Q (q1, q2)
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12. Example:
Find the component form and magnitude of the vector v with initial point P = (3, 2) and terminal point Q = (1, 1). =
, 3 4 -
p1 , p2 = 3, 2 q1 , q2 = 1, 1 So, v1 = 1  3 =  4 and v2 = 1  ( 2) = 3. Therefore, the component form of v is
, v2 v1
The magnitude of v is
||v|| = = = 5.
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Example: Magnitude

13. Example: Equal Vectors
Two vectors u = u1, u2 and v = v1, v2 are equal if and only if u1 = v1 and u2 = v2 .
Example: Equal Vectors
Example: If u = PQ, v = RS, and w = TU with P = (1, 2), Q = (4, 3), R = (1, 1), S = (3, 2), T = (-1, -2), and U = (1, -1), determine which of u, v, and w are equal.
Calculate the component form for each vector:
u = 4  1, 3  2 = 3, 1
v = 3  1, 2  1 = 2, 1
w = 1  (-1), 1  (-2) = 2, 1
Therefore v = w but v = u and w = u. /
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14. Operations on Vectors in the Coordinate Plane
Let u = (x1, y1), v = (x2, y2), and let c be a scalar.
1. Scalar multiplication cu = (cx1, cy1)
2. Addition u + v = (x1+x2, y1+ y2)
3. Subtraction u  v = (x1  x2, y1  y2)
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15. Examples: Operations on Vectors
Examples: Given vectors u = (4, 2) and v = (2, 5) x y
-2u = -2(4, 2) = (-8, -4)
(4, 2) u 2u
(-8, -4)
u + v = (4, 2) + (2, 5) = (6, 7)
u  v = (4, 2)  (2, 5) = (2, -3) x y x y
(2, 5)
(4, 2) v u
(6, 7)
(2, 5)
(4, 2) v u
u + v
u  v
(2, -3)
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Examples: Operations on Vectors

16. Direction Angle
The direction angle  of a vector v is the angle formed by the positive half of the x-axis and the ray along which v lies. x y x y θ v v θ
If v = x, y , then tan  = . x y v
(x, y)
If v = 3, 4 ,
then tan  = and  = 51.13.
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