Vectors a vector measure has both magnitude (size) and direction.

Some Vector & Scalars: Vectors Scalars Displacement distance Velocity speed Acceleration temperature Force time Momentum mass All fields.

Concurrent Vectors

Sequential Vectors

Combining (adding or subtracting) Vectors: Mathematical operations can be done on non-linear vectors – but not in the usual way. Their direction has to be taken into account. We cannot simply add 40 km North + 20 km East. The resulting displacement is not 60 km.

Methods for vector addition when vectors are not in a straight line Methods for vector addition when vectors are not in a straight line. 1) Vectors not at 90o, use graphical analysis (a scaled diagram) tail to tip. 2) For 2 vectors at 90o to each other use the Pythagorean theorem to solve.

Vectors not at right angles Vectors not at right angles. Sketch a scaled vector diagram using one of two methods: 1) Tail to tip 2) Parallelogram (Two vectors only) Graphical Analysis

Negative vectors - opposite positive ones. 10 m/s East = +10 m/s West 36 km 20o N of E = +36 km 20o S of W. What does –10 m South mean? 10 m North

Subtraction: Reverse the direction of the negative vector & add graphically (make your scaled diagram). 12 km East – 6 km south 12 km East + 6 km north. 13 m/s north – 5 m/s 20o N of E = 13 m/s north +5 m/s 20o S of W

Equilibrant is a vector that “neutralizes” the resultant Equilibrant is a vector that “neutralizes” the resultant. It is equal and opposite the resultant. Ex: R = 25 m/s South, Equilibrant = 25 m/s North or (-25m/s S)

R E

Note: the tail to tip vector diagram may be to resolve any components more than two. The parallelogram method may be used to resolve only two vector components. The Pythagorean theorem may only be used for vectors at right angles.

Resolution of Resultant to Components All 2-d vectors can be described as the sum of perpendicular vectors.

Vector a can be broken down, or resolved into 2 perpendicular components: ay & ax.

sin q = ay cos = ax. a a ay = a sin q . ax = a cos q .

Finding Resultant Algebraically To find resultant of 2 or more vectors, we can resolve each vector into the X and Y components. Then we can add the x components & Y components separately & reconstruct the resultant vector.

Find the resultant of the 2 vectors below:

Each vector can be resolved to X & Y components.

The X & Y components can be added because they are in a straight line Find the resultant from Pythagorean.

1. A hiker walks 25. 5 km from her base camp at 35o S of E 1. A hiker walks 25.5 km from her base camp at 35o S of E. On the second day she walks 41 km in a direction 65o N of E. Determine the magnitude and direction of the resultant displacement. 1. Use algebraic methods. 2. Use scaled vector diagram.

Component Vectors

Example Problem Kerr pg 27 #6 – 8. Good 12 min Youtube Vector Component Lesson https://www.youtube.com/watch?v=tvrynGECJ7k

Free Body Diagrams. Show Vector Forces as Arrows.