Mathematics with vectors When dealing with more than one dimension, can have the same equation come up multiple times. Example: Position equation in three dimensions yields

In a situation with multiple objects and multiple physical quantities, the number of equations increases dramatically! –Need a shorthand to simplify the process. –Vectors provide this simplification. Note that vectors provide a shorthand to see physical relationships, but never allow for numerical answers!

Definition of vectors Vectors are a set of numbers that describe the same physical quantity –Number of numbers equals the number of dimensions Example: Blood pressure – use both systolic and diastolic to determine health Vectors are physical objects with magnitude and direction Example: Herd migration – number of animals and direction of movement determines migration.

Components Magnitude of vector depicted by length Direction by angle Components are projection onto the various axes –Length along axis is magnitude in that direction –One component for each axis

Shorthand Notation is shorthand for where letter subscripts indicate projection axis

Addition/Subtraction Add and subtract vectors by components –Notice that if B has a component in an opposite direction of A, then components subtract. Subtract vectors by reversing direction of subtracted vector and adding –Reverse direction by changing sign on all components.

Vector Multiplication Three types of vector multiplication –scalar x vector = vector –vector x vector = scalar –vector x vector = vector Scalar (number) x vector simply “stretches” vector –Scalar multiplies each component equally

Scalar (Dot) Product vector x vector = scalar –Product of magnitude of one vector times projected magnitude of other

Vector (Cross) Product vector x vector = vector –Two (perpendicular) vectors define plane –Direction of product is at right angles to this plane –Magnitude is product of perpendicular vectors

Ballistic Motion Ballistic motion is the most basic motion in two dimensions. –Only acceleration is that of gravity –Only acts in one direction –Motion is in two dimensions

1.at the point just before the projectile lands 2.at the highest point 3.at the launch point 4.nowhere As a projectile thrown upward at a non-vertical angle moves in a parabolic path, at what point along its path are the velocity and acceleration vectors for the projectile parallel to each other?

1.at the point just before the projectile lands 2.at the highest point 3.at the launch point 4.nowhere As a projectile thrown upward at a non-vertical angle moves in a parabolic path, at what point along its path are the velocity and acceleration vectors for the projectile parallel to each other?

Solving ballistic motion problems Treat each direction independently Example: A field biologist is collecting specimens. She spots a rare monkey 10 m up in a tree 35 m away.Moving carefully, she fires a tranquilizer dart at the monkey. Unfortunately, at the moment the trigger is pulled, the monkey lets go. If the dart leaves the gun at 45 m/s, does the dart hit the monkey?

1) Find initial angle Using trigonometry, get

2. Find time for dart to travel horizontally to tree Apply position equation to horizontal motion

3. Find dart’s distance above ground when it reaches tree Use distance equation for vertical motion 4. Find distance of monkey above ground

As a projectile thrown upward at a non-vertical angle moves in a parabolic path, at what point along its path are the velocity and acceleration vectors for the projectile perpendicular to each other? 1.at the point midway between the launch point and the highest point 2.at the launch point 3.nowhere 4.at the highest point