Vectors

Overview

(Holt p82)

Motion Progression (1D 2D)

Started with 1 dimensional (1D) motion • Motion along X or Y axis

Now consider 2 dimensional motion (2D), that is between the Y and X axes • Resulting motion is a VECTOR (V)

that can be resolved (broken down) into x (Vx) and y (Vy) components, as shown, where V is the size and θ the angle of the vector.

Y

X

Vx

Vy

V

Geometry background

Remember the measurement baseline

from geometry

Quadrants I, II, III, IV

What is a vector?

A vector is a quantity that is represented by its magnitude (size/value) and its direction: • Its magnitude is represented by the length of

a straight line with an arrow head showing…

• its direction, represented by the angle the line makes with a reference grid (usually the + x-axis)

• Example: a velocity of 20 m/s NE

Examples of vectors (vs scalars)

Displacement (vs distance)

Velocity (vs speed)

Acceleration

Force

Momentum

Questions

Which of the following sentences deal

with vector quantities?

• I used to drive a 10-ton truck.

• You’ll find a gas station if you drive 20 miles

north.

• I have been sailing for 20 km and still have

not seen the island.

• The 10-volt battery on your left is dead.

Vectors in Daily Use

Navigation

• Airplanes

• Sailboats

Applications of multiple forces

Head/Tail wind

Cross wind

What can we do with vectors?

Arithmetic add/subtract

• Graphical methods

• Tip-tail

• Parallelogram

A

B

C = resultant

A

B

C = resultant

What can we do with vectors?

Arithmetic add/subtracting vectors • Analytical methods

• Resolve each vector (V) into its x & y components

• Vx = V cos

• Vy = V sin

• Then add all the x components and all the y components to get the resultant in terms of x & y

• The magnitude of the resultant is “Pythagorean”

• (Vx2 + Vy

2) -> c2 = a2 + b2

• The direction of the resultant () is (trigonometrically)

• tan-1(Vy/Vx)

• Remember SOH CAH TOA

Practice - Navigation

A plane leaves Weyers Cave and flies north for 50km before turning east for 60km. What is the plane’s final position (x km @ θ degrees) from its start point?

Solve: 1) distance and 2) direction • 1) Pythagorean Theorem

• x2 = 502 + 602

• x = 78.1 km

• 2) Trigonometry • tan θ = opp/adj = 50/60

• θ = tan-1 (50/60) = 39.8º

Answer: 78.1 km @ 39.8º

50 km

60 km

θ

x

Resolving Vectors Practice

Relative direction • Ex. 25 N of W

Multi-vector models • Resolve each vector into x & y components

remembering the +,- values in a coordinate plane

• Add the x’s and the y’s

• Use Pythagoras to get the distance from the x, y additions

• Use tan-1 to get horizontal angle

Additional Math tools

What if the path is NOT right angle based? • Construct right triangles and use Pythagoras

• SOH CAH TOA

• Use trig tools • Law of Sines

• (sin A)/a = (sin B)/b = (sin C)/c

• Law of Cosines

• a2 = b2 + c2 - 2bc cosA

• b2 = c2 + a2 – 2ca cos B

• c2 = a2 + b2 – 2ab cos C

Practice

A boat sails due north for 40 kms before

realizing that the dog has fallen overboard,

requiring a southern trip of 10 kms to retrieve

the sodden canine! The boat resumed its

northerly sail for 40 kms before heading west

for 60 kms.

• Draw the boat’s path and determine where the

boat is, relative to its start point? (vector =

length + direction, required)

Solution

North vector = 40-10+40 = 70 km

West vector = 60 km

Resultant (using Pythagoras)

• x2 = 702 + 602 = 8500

• x = 92.19 km

Angle

• Ө = tan-1(opp/adj) = tan-1 (60/70)

• Ө = 40.6º West of North

Ө

x

Practice – Your Turn

A plane flies north for 50 km, west for 40

km, then SW for 20 km before sending

out a “mayday” call.

• Where is the plane with respect to its start

point?

Graphical Vector Resolution

“Box in” the vector (V) with a rectangle

Read off the horizontal (x) value = Vx

Read off the vertical (y) value = Vy

Measure the horizontal angle and

describe it uniquely

Vx

Vy

Projectile Motion

Projectile Motion