4-1 Properties of Vectors

The line and arrow used in Ch.3 showing magnitude and direction is known as the Graphical Representation◦ Used when drawing vector diagrams

When using printed materials, it is known as Algebraic Representation◦ Italicized letter in boldface◦d = 50 km SW

Representing Vector Quatities

Two displacements are equal when the two distances and directions are equal◦ A and B are equal, even though they don’t begin

or end at the same place

This property of vectors makes it possible to movevectors graphically for adding or subtracting

Resultant Vectors

A

Vectors shown are unequal, even though they start at the same place

◦ C

Resultant Vectors

D

The resultant vector is the displacement of the vector additions.

My route to school is My resultant vector is R 0.50 miles East 2.0 miles North 2.5 miles East 20.0 miles North 2.5 miles East Resultant Vector = 23 miles NE

Resultant Vector

R

When manipulating graphical reps. of vectors, need a ruler to measure correct length

Take the tail end and place at the head of the arrow◦ Enroute to a school, someone travels 1.0 km W,

2.0 km S, and then 3.0 km W◦ Resultant vector =

4.5 km SW

Graphical Addition of Vector

Vectors added at right angels can use the Pythagorean System to find magnitude

If vectors added and angle is something other than 90o, use the Law of Cosines◦ R2 = A2 + B2 – 2ABcosθ

Magnitude of the Resultant

Find the magnitude of the sum of a 15 km displacement and a 25 km displacement when the angle between them is 135o.◦ A = 15 km; B = 25 km; θ = 135o; R = unknown◦ R2 = A2 + B2 – 2ABcosθ◦ = (25 km)2 + (15 km)2 – 2(25km)(15 km)cos135o

◦ =625 km2 + 225 km2 – 750km2(-0.707)◦ =1380 km2

◦ R = √1380km2

◦ = 37 km

Magnitude of the Resultant

A hiker walks 4.5 km in one direction, then makes a 45o turn to the right and walks another 6.4 km. What is the magnitude of her displacement?

Problem

A person walked 450.0 m North. The person then turned left 65o and traveled 250.0 meters. Find the resultant vector.

Multiplying a Vector by a scalar number changes its length, but not direction, unless negative◦ Vector direction is then reversed◦ To subtract two vectors, reverse direction of the

2nd vector then add them◦ Δv = v2 – v1

◦ Δv = v2 + (-v1)

◦ If v1 is multiplied by -1, the direction of v1 is reversed and can be added to v2 to get Δv

Subtracting Vectors

Graphical addition can be used when solving problems that involve relative velocity◦ School bus traveling at a velocity of 8 m/s. You

walk toward the front at 3 m/s. How fast are you moving relative to the street?

◦ vbus relative to street

◦ vyou relative to bus

◦ vyou relative to the street

Relative Velocities

When a coordinate system is moving, two velocities add if both moving in the same direction & subtract if the motions are in opposite directions◦ What if you use the same velocities and walk to the rear

of the bus. What is your resultant velocity relative to the street?

◦ vbus relative to the street

◦ vyou relative to the bus

◦ vyou relative to the street

Relative Velocities

Suppose an airplane pilot wants to fly from the U.S. to Europe. Does the pilot aim the plane straight to Europe?◦ No, must take in consideration for wind velocity

v air relative to the ground

v plane relative to air

v plane relative to ground

Relative Velocities in 2-Dimensions