Vectors In A Single Plane

Vector Representation Have you ever drawn a treasure map as a child? Have you ever drawn a treasure map as a child? Drawn a map to you home for someone else? Drawn a map to you home for someone else? Vector quantities are represented by arrows that point in the direction of the quantity Vector quantities are represented by arrows that point in the direction of the quantity

Arrows for Vectors The length of the arrow – magnitude of quantity ( drawn to scale) The length of the arrow – magnitude of quantity ( drawn to scale) The direction of the arrow – direction of quantity (reference point) – you need a coordinate system or frame of reference The direction of the arrow – direction of quantity (reference point) – you need a coordinate system or frame of reference(diagram)

Three ways to indicate direction of vectors Three ways to indicate direction of vectors Angles degrees Angles degrees NSEW [N 30° E] NSEW [N 30° E] Bearings Bearings

Adding Vectors Can’t add apples and oranges Can’t add apples and oranges You can only add vectors that represent the same quantity and are drawn with the same scale (displacements, forces) You can only add vectors that represent the same quantity and are drawn with the same scale (displacements, forces) Resultant – sum of all vectors Resultant – sum of all vectors

Steps for Adding Vectors Set up coordinate system Set up coordinate system Place vector A Place vector A Place the tail of vector B at the tip of vector A (tip to tail) Place the tail of vector B at the tip of vector A (tip to tail) Repeat step 3 if more than one vector Repeat step 3 if more than one vector Draw a vector from the tail of the first vector to the tip of the last vector. Label this as you resultant Draw a vector from the tail of the first vector to the tip of the last vector. Label this as you resultant Use a ruler to measure the length of the resultant Use a ruler to measure the length of the resultant Use a protractor to measure the angle between the resultant and the horizontal axis Use a protractor to measure the angle between the resultant and the horizontal axis

Subtracting Vectors Subtracting Vectors graphically Subtracting Vectors graphically Δd = d 2 – d 1 Δd = d 2 – d 1 Δv = v 2 – v 1 Δv = v 2 – v 1 A-B same as A + (-B) A-B same as A + (-B) Negative vectors has the same magnitude and opposite direction Negative vectors has the same magnitude and opposite direction

Steps for Subtracting Vectors Set up coordinate system Set up coordinate system Place vector A Place vector A Place the tail of negative vector B at the tip of vector A (tip to tail) Place the tail of negative vector B at the tip of vector A (tip to tail) Draw a vector from the tail of the first vector to the tip of the last vector. Label this as you resultant Draw a vector from the tail of the first vector to the tip of the last vector. Label this as you resultant Use a ruler to measure the length of the resultant (magnitude) Use a ruler to measure the length of the resultant (magnitude) Use a protractor to measure the angle between the resultant and the horizontal axis Use a protractor to measure the angle between the resultant and the horizontal axis

Multiplying and Dividing Vectors What happens to a vector when it is multiplied or divided by a scalar value? What happens to a vector when it is multiplied or divided by a scalar value? V = Δd / t V = Δd / t When displacement (vector) is divided by time (scalar) the resulting vector has a new magnitude and unit but the direction remain the same. When displacement (vector) is divided by time (scalar) the resulting vector has a new magnitude and unit but the direction remain the same.

Relative Velocity Analyze quantitatively, the motion of an object is relative to different reference points Analyze quantitatively, the motion of an object is relative to different reference points Eg: stopped at a red light, some times it feels like you’re moving backwards…are you? Eg: stopped at a red light, some times it feels like you’re moving backwards…are you? Vector addition is a critical tool in calculating relative velocities Vector addition is a critical tool in calculating relative velocities

Pilot – ground and air as frames of reference Pilot – ground and air as frames of reference You must account for both the motion of the plane relative to the air and the air relative to the ground You must account for both the motion of the plane relative to the air and the air relative to the ground Apply velocity vectors for each Apply velocity vectors for each Plane to air velocity Plane to air velocity + air to ground velocity + air to ground velocity plane velocity to ground plane velocity to ground