1. 6.3-Vectors in the Plane

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 physical quantities.

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 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
Directed Line Segment

4. 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

5. Geometric- Vector Additionu
To add vectors u + v:
1. Place the initial point of u at the terminal point of v.
2. Draw the vector with the same initial point as v and the same terminal point as u. v u u v
u + v

6. To subtract vectors u - v:
Vector Subtraction
Vector Subtraction v u
To subtract vectors u - v:
1. Place the initial point of -v at the terminal point of u.
2. Draw the vector u  v from the terminal point of v to the terminal point of u. u
u  v -v

7. 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)

8. 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.
Example: Magnitude

9. 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. /

10. 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)

11. 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)
Examples: Operations on Vectors

12. Unit Vectors
Simply put, a unit vector is a vector which has a length of one unit.
is a unit vector because its magnitude is:
is a unit vector because its magnitude is:

13. A unit vector can be found by dividing a given vector by its magnitude:
Ex. Find a unit vector in the direction of v = <-2, 5>

14. are special standard unit vectors in the
positive X, Y direction.
Ex. 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.

15. Ex. Let and Find 2u – 3v.

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 x y v
(x, y) θ

17. Ex. If v = <3, 4>, what is the direction angle? y v
(x, y)
How could we determine the angle? θ
If v = x, y , then tan 
Ex. If v = <3, 4>, what is the direction angle?
tan  = and  = 51.13