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Chapter 2 - Vectors
Objectives• Understand vectors and their
components on the coordinate system
• Find the magnitude of a vector• Understand vector addition by
the parallelogram method and by the component method
• Understand vectors in a state of equilibrium
Vectors on the Coordinate System
• Horizontal, vertical, and slanted vectors can be drawn on the coordinate system.
• All 3 types of vectors have both length and direction.
Slanted Vectors
• The direction of slanted vectors is stated in terms of – The angle formed by the vector and the
horizontal axis. – The quadrant in which that angle is
formed.
• The length of this vector is ___?• The direction of this vector is a ___
angle in the ___ quadrant.3 units
Θ = 40˚
Slanted Vectors
• The angle which specifies the direction of a slanted vector is called its reference angle.
• All slanted vectors have positive lengths.
• Vectors are named using 2 letters:– AB
• The first letter of the name is always where the vector begins.
3 units
Θ = 40˚A
B
Slanted Vectors
• Any slanted vector has a horizontal and a vertical component.
• We can calculate these because we can make this a right triangle and use trig.
3 units
Θ = 40˚A
B
Magnitude
• How long is this vector?• Use the distance formula!
• If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is
• AB = √(x2 - x1)2 + (y2 - y1)2
• AB = (4-0)2 + (5-0)2
• AB = √ 16 + 25• AB = √41
4,5
0,0A
B
Magnitude
• Try another one:
• AB = √(x2 - x1)2 + (y2 - y1)2
• AB = √(5-0)2 + (4-2)2
• AB2 = √ 25 + 4• AB = √29 = 5.4
5,4
0,2 A
B
Component Form
• The component form of a vector is written as <x, y> where x is (x2 - x1) and y is y2 – y1
• What is the component form of this vector?
• <(5-0), (4-2)>• <5, 2>
5,4
0,2 A
B
Another example
• What is the component form of this vector?
• <4, 5>
4,5
0,0A
B
Direction
• The direction of a vector is determined by the angle it makes with the horizontal line.
• What direction is vector AB heading?• If AB represents the velocity of a
moving ship, and the scale on the axis is miles per hour, how fast is the ship moving? 3,4
0,0A
B
Equal and Parallel Vectors
• Two vectors are equal if they have the same magnitude and direction.
• Two vectors are parallel if they have the same or opposite directions.
3,4
0,0A
B
Slanted Vectors
• How do we calculate the horizontal component (AC)?
• Cos θ = adj/hyp = x/3• .7660 = x/3• X = 3 * .7660 = 2.298
• sin θ = opp/hyp = x/3• .6428 = x/3• X = 3 * .7660 = 1.9284
• Use Pyth to check• 2.2982 + 1.92842 ?=? 32
3 units
Θ = 40˚A
B
C
Flipping the problem
• Tan = opp/adj• Tan θ = 4/5 = .8000• Therefore θ contains 39˚
• Pyth can help us find the length of AB:• AB2 = AC2 +BC2 • AB = 52 + 42
• AB = 25 + 16 = 41 • AB = 6.4
• How would you do this using sin and cos?
Θ = ?˚A
B = (5, 4)
C
Adding Vectors• What does it mean to add two
vectors?• Vector and Field (vector addition)• Why do we care? Using Vectors
Video
A= 5,20
C = -4,3θ
Adding Vectors
• In Physics, the Law of Conservation and Momentum uses this.
• Now how do we do that without the website?
• Create a parallelogram and find the diagonal.
A= 5,20
C = -4,3θ
Adding Vectors• Draw AQ which is both parallel
to OC and equal in length to OC.
• Draw CQ which is both parallel to OA and equal in length to OA
• On a graph, we can see that the points of Q are 2,6
A= 5,20
C = -4,3θ
Q
Adding Vectors• We can draw one line, then a vector
from the origin to point Q:• This lets us find the point on graph
paper without a calculator. Even using a calculator, this is a nice way to prove we’re doing things correctly.
• Would this be precise if we weren’t using whole numbers?
A= 5,20
C = -4,3θ
Q
Adding Vectors• Another way to add vectors is
by the component method.– This provides accurate answers
without the necessity of constructing parallelograms.
• Find the horizontal and vertical components, and add them
• Horizontal: -3 + 5 =2• Vertical: 4 + 2 = 6
A= 5,20
C = -3, 4θ
Q
Vector addition• Positives and negatives are
extremely important – be careful with them.
A= 5,20
C = -4,3θ
Q
Vector addition• To find the length and direction
of the resultant vector, we use trig.
• Use Pyth to find the length of OC
• Use tan to find the reference angle of OC
C= 25.6, 12.70
F = -7.9, 7.2
α
Q
Applications
• How is vector addition used in physics?
• Law of Conservation Video
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