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SP1b. Compare and constract scalar and vector quantities.
Trigonometry and Vectors
Motion and Forces in Two Dimensions
SP1b. Compare and constract scalar and vector quantities.
Standard
SP1. Students will analyze the relationships between force, mass, gravity, and the motion of objects.
b. Compare and contrast scalar and vector quantities.
SP1b. Compare and constract scalar and vector quantities.
Right Triangles
• The longest side is the hypotenuse. It is opposite the 90º angle.
• The other two sides are named depending on where they are in relation to the angle you are looking at.
SP1b. Compare and constract scalar and vector quantities.
The Pythagorean Theorem
• For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.
SP1b. Compare and constract scalar and vector quantities.
Animation
SP1b. Compare and constract scalar and vector quantities.
Formula
• The formula we use is:a2 + b2 = c2
• C = the length of the hypotenuse• The other two sides are a and b.
SP1b. Compare and constract scalar and vector quantities.
We Also Need to Know the Angles• Acute angle A is drawn in
standard position as shown.
Right-Triangle-Based Definitions of Trigonometric FunctionsFor any acute angle A in standard position,
adjacent side
opposite sidetan
hypotenuse
adjacent sidecos
hypotenuse
opposite sidesin
x
yA
r
xA
r
yA
opposite side
adjacent sidecot
adjacent side
hypotenusesec
opposite side
hypotenusecsc
y
xA
x
rA
y
rA
SP1b. Compare and constract scalar and vector quantities.
Easy Way To Remember
Example Find the values of sin A, cos A, and tan A
in the right triangle.
Solution– length of side opposite angle A is 7– length of side adjacent angle A is 24– length of hypotenuse is 25
247
tan,2524
cos,257
sin AAA
SP1b. Compare and constract scalar and vector quantities.
VECTORS AND VECTOR RESOLUTION
SP1b. Compare and constract scalar and vector quantities.
Scalar
SP1b. Compare and constract scalar and vector quantities.
Vector
SP1b. Compare and constract scalar and vector quantities.
Vectors
SP1b. Compare and constract scalar and vector quantities.
Vector Addition
• VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them.
• Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started.
SP1b. Compare and constract scalar and vector quantities.
Vector Subtraction
• VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT.
• Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started.
SP1b. Compare and constract scalar and vector quantities.
More Examples
SP1b. Compare and constract scalar and vector quantities.
Angle Direction
SP1b. Compare and constract scalar and vector quantities.
Vectors Are Typically Drawn to Scale
SP1b. Compare and constract scalar and vector quantities.
So How Do We Add These?
SP1b. Compare and constract scalar and vector quantities.
PYTHAGOREAN THEOREM
SP1b. Compare and constract scalar and vector quantities.
Example
SP1b. Compare and constract scalar and vector quantities.
Example
• Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.
SP1b. Compare and constract scalar and vector quantities.
PARALLELOGRAM METHOD
SP1b. Compare and constract scalar and vector quantities.
You Might Find this Useful
SP1b. Compare and constract scalar and vector quantities.
USING TRIG FUNCTIONSDetermining Direction
Vectors can be broken down into components
A fancy way of saying how far it goes on the x and on the y.
We calculate the sides using trig.
SP1b. Compare and constract scalar and vector quantities.
"A" is used to represent the vector.
cos
cos
cos
x
x
A
HA
AA A
SP1b. Compare and constract scalar and vector quantities.
Now the Other Side
sin
sin
sin
y
y
O
HA
AA A
SP1b. Compare and constract scalar and vector quantities.
Example
• What if you have a vector that is 45 m @ 25o?
cos
45cos 25
40.8
x
x
x
A A
A
A m
sin
45sin 25
19.0
y
y
y
A A
A
A m
SP1b. Compare and constract scalar and vector quantities.
SP1b. Compare and constract scalar and vector quantities.
So How Do We Find Value of Direction?
• The direction is given as an angle from the +x axis
• Positive is counterclockwise
1
tan
tan
tan
y
x
y
x
O
AA
A
A
A
SP1b. Compare and constract scalar and vector quantities.
GRAPHICAL METHOD (TIP TO TAIL)
SP1b. Compare and constract scalar and vector quantities.
Graphical Method
SP1b. Compare and constract scalar and vector quantities.
The Order Doesn’t Matter
• Same three vectors, different order
SP1b. Compare and constract scalar and vector quantities.
Animation
SP1b. Compare and constract scalar and vector quantities.
Resultants
SP1b. Compare and constract scalar and vector quantities.
SP1b. Compare and constract scalar and vector quantities.
SP1b. Compare and constract scalar and vector quantities.
SP1b. Compare and constract scalar and vector quantities.
• (100 km/hr)2 + (25 km/hr)2 = R2
• 10000 km2/hr2 + 625 km2/hr2 = R2
• 10625 km2/hr2 = R2
• SQRT(10 625 km2/hr2) = R• 103.1 km/hr = R
SP1b. Compare and constract scalar and vector quantities.
• tan (theta) = (opposite/adjacent)• tan (theta) = (25/100)• theta = invtan (25/100)• theta = 14.0 degrees
SP1b. Compare and constract scalar and vector quantities.
Animation
SP1b. Compare and constract scalar and vector quantities.
Steps in vector addition
1. Sketch the vector– Head to tail method
– Parallelogram method
2. Break vectors into their components
3. Add the components to calculate the components of the resultant vector
4. Calculate the magnitude of R
5. Calculate the direction of R
Add 180o to q if Rx is negative
SP1b. Compare and constract scalar and vector quantities.
Example• A boat moves with a velocity of 15 m/s, N in a
river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north.
SP1b. Compare and constract scalar and vector quantities.
Example• A plane moves with a velocity of 63.5 m/s at
32 degrees South of East. Calculate the plane's horizontal and vertical velocity components.
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