Vectors Measured values with a direction Shown with +/- in each dimension Shown as arrow for two-dimensions

Vectors Measured values with a direction Shown with +/- in each dimension Shown as arrow for two-dimensions Length of arrow represents the size or magnitude of the measurement 5 m/s would be shown with a shorter arrow than 20 m/s

Vectors These may only be used for Right Triangle (a triangle with a right angle)

Vectors The legs are the sides of the triangle that are next to the right angle The hypotenuse is across from the right angle It is always the longest side of the triangle

Vectors Hypotenuse Leg Leg

Vectors c a b

Vectors Pythagorean Theorem – used to find the length of each side of a right triangle a2 + b2 = c2 c a b

Trig Functions If another angle ( θ ) is known, trig functions can be used c a b θ

Vectors c 3 4 10 6 x

North: South: East: West:

North: up South: down East: right West: left

Horizontal vectors Vertical vectors

x-value Horizontal vectors: X-direction Vertical vectors: Y-direction y-value

Vector addition Vector resolution combining two (or more) vectors to produce a single resultant vector Vector resolution showing a single vector as a combination of other vectors (using vector addition) we will find horizontal and vertical to be the most beneficial components to use

As long as both direction (angle) and size (magnitude) of the vector are not changed, they can be moved and rearranged as needed To show addition, all vectors must be aligned “tip-to-tail”

Draw triangles and use trig equations to solve for both vector addition and vector resolution

Examples Vector Addition Show the x-value arrow, then the y-value arrow Remember: tip-to-tail Then show the single arrow that does the same thing Use Pythagorean Theorem to solve

Combine the following vectors: 300 meters south 400 meters west

Combine the following vectors: 300 meters south 400 meters west 400 m

Combine the following vectors: 300 meters south 400 meters west 400 m 300 m

Combine the following vectors: 300 meters south 400 meters west 400 m 400 300 m vT

Combine the following vectors: 300 meters south 400 meters west vT Find vT

a2 + b2 = c2 3002 + 4002 = vT2 vT = 500 meters 400 m 300 m vT

a2 + b2 = c2 3002 + 4002 = vT2 vT = 500 meters 400 m 300 m 500 meters

Examples Vector Resolution Show the single v-arrow and angle with a dotted line Then show the x-arrow and y-arrow that combine tip-to-tail to make up the single v arrow

Vector resolution θ

Vector resolution θ vx

Vector resolution vy θ vx

Example

Show the horizontal and vertical components of the following vector: 100 m/s @ 30° 30° 100 m/s

Show the horizontal and vertical components of the following vector: 100 m/s @ 30° x 30° 100 m/s

Show the horizontal and vertical components of the following vector: 100 m/s @ 30° x 30° y 100 m/s

Special Triangles a 45° b

Special Triangles Legs are same length with 45° triangle a = b a 45° b

Special Triangles Legs are same length with 45° triangle a = b a 45°

Special Triangles Legs are same length with 45° triangle a2 + b2 = c2 a2 + a2 = c2 a 45° b a

Special Triangles Legs are same length with 45° triangle a2 + a2 = c2 b a

Complimentary Angles Two angles that add up to 90°

Complimentary Angles Two angles that add up to 90° The two angles inside a triangle are complimentary

Complimentary Angles Two angles that add up to 90° The two angles inside a triangle are complimentary When the angles inside are switched, the legs of the right triangle are also switched

Complimentary Angles a 30° b

Complimentary Angles 60° a 30° b

Complimentary Angles 60° a 30° b 60°

Complimentary Angles 60° a 30° b 30° 30 and 60 are complimentary 60°

Complimentary Angles a 30° b b 60° a