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: VectorsScalars Displacementdistance Velocityspeed Accelerationtemperature Forcetime Momentummass All field values.

Concurrent Vectors R must fall between Parallelogram works well

Combining (adding) Vectors: Graphical Methods

Vectors not at right angles. Sketch a scaled vector diagram using one of two methods: Parallelogram (Two vectors only) Tail to tip 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 30 o N of E. What is the final velocity? 6.5 m/s 4.5 m/s 30 o

If only 2 vectors. Turn your two vector components into a parallelogram. Parallelogram Method 30 o 4.5 m/s 6.5 m/s

6.0 m/s 4.5 m/s 35 o When the parallelogram is complete, sketch the resultant between the two original component vectors corner to corner. Measure the resultant and the new angle.

Tail to tip method.

Tail to Tip. 6.5 m/s 4.5 m/s 30 o

Tail to Tip: Sketch a diagram with a scale of 1cm = 1 m/s. Sketch each vector separately 1 at a time. Place the tail of one of the vectors to the tip of the other. 6.5 cm 4.5 cm 30 o

Now connect a straight line, the resultant, from the tail of the unmoved (1st)vector to the tip of the 2nd vector (moved). 6.5 cm 4.5 cm 30 o R

Measure the resultant with your ruler to get the magnitude. Measure the angle to get the direction of the resultant. R=10.6 m/s  =13 o

Negative vectors are in the opposite direction of positive ones. -10 m/s East = +10 m/s West - 36 km 20 o N of E = +36 km 20 o 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 20 o N of E = 13 m/s north +5 m/s 20 o S of W

Changes in velocity  v, momentum  p etc. Means subtract the final. A pool ball traveling southeast strikes the bumper of the table at a 30 o angle to perpendicular at 6 m/s. It rebounds southwest at the same 30 o angle and speed. Find the change in velocity.  v = v f - v i.

Equilibrant is a vector that “neutralizes” the resultant. It is like subtraction! 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: a y & a x.

a y = a sin . a x = 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 A crate is being dragged by students who pull on two ropes. One rope is pulled 40 o N of E with a force of 45- N, the other is pulled 25 o S of E with a force of 30-N. Use algebraic methods to find the resultant force vector and direction. Kerr pg 27 #3, 5 – 8.

Free Body Diagrams. Show Vector Forces as Arrows.