Vectors

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

The symbol for a vector is a letter with an arrow over it or boldface type V.

Some Vector & Scalars:

Vectors ScalarsDisplacement distanceVelocity speedAcceleration temperatureForce timeMomentum massAll fields.

Combining (adding) 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.

1) Vectors not at 90o, use graphical analysis (a diagram).

2) For 2 vectors at 90o to each other use the Pythagorean theorem to solve.

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

Two people kick a ball at the same time. One gives it a velocity of 6.5 m/s east, the other gives it a velocity of 4.5 m/s 30o N of E. What is the final velocity?

6.5 m/s

4.5 m/s

30o

If only 2 vectors. Turn your two vector components into a parallelogram.

Parallelogram Method

30o

4.5 m/s

6.5 m/s

6.0 m/s

4.5 m/s

35o

When the parallelogram is complete, sketch the resultant between the two original component vectors corner to corner. Measure the resultant and the new angle.

Negative vectors are in the opposite direction of 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: Just 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.

It is equal and opposite the resultant.

Ex: R = 25 m/s South, Equilibrant = 25 m/s North or (-25m/s S)

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.

Instead of combining vector components to give resultant, we take resultant &resolve it (break it ) into perpendicular components.

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

ay = a sin . ax = a cos .

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:

To find the resultant.

Example Problem

• Kerr pg 27 #6 – 8.

Free Body Diagrams.

• Show Vector Forces as Arrows.