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This guide explores vector representation in a single plane, drawing parallels to treasure maps and home communication. Vectors are illustrated as arrows indicating magnitude and direction, employing a coordinate system for clarity. It outlines methods for adding and subtracting vectors, explaining the concept of resultants and negative vectors graphically. Furthermore, the multiplication and division of vectors by scalar values are discussed. Finally, it emphasizes the importance of vector addition in analyzing relative velocities in different frames of reference, crucial for applications like aviation.
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Vectors In A Single Plane
Vector Representation • Have you ever drawn a treasure map as a child? • Drawn a map to you home for someone else? • 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 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 • Angles 0-360 degrees • NSEW [N 30° E] • Bearings
Adding Vectors • 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) • Resultant – sum of all vectors
Steps for Adding Vectors • Set up coordinate system • Place vector A • Place the tail of vector B at the tip of vector A (tip to tail) • 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 • Use a ruler to measure the length of the resultant • Use a protractor to measure the angle between the resultant and the horizontal axis
Subtracting Vectors • Subtracting Vectors graphically • Δd = d2 – d1 • Δv = v2 – v1 • A-B same as A + (-B) • Negative vectors has the same magnitude and opposite direction
Steps for Subtracting Vectors • Set up coordinate system • Place vector A • 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 • 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
Multiplying and Dividing Vectors • What happens to a vector when it is multiplied or divided by a scalar value? • 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.
Relative Velocity • 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? • Vector addition is a critical tool in calculating relative velocities
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 • Apply velocity vectors for each • Plane to air velocity + air to ground velocity plane velocity to ground
