Vectors

VECTOR: has magnitude as well as direction Vectors and Scalars VECTOR: has magnitude as well as direction examples: displacement, velocity, force, momentum SCALAR: has only a magnitude examples: mass, time, temperature

Vector Directions E (0o) N (90o) W (180o) S (270o)

Vectors and Scalars 60o 60o 60o North of East 30o East of North You can describe the direction of the following vector in three ways: 60o 60o 60o North of East 30o East of North

Adding Vectors – Graphical RESULTANT: the single vector obtained by adding or combining two or more vectors (components) The resultant of any two vectors can be found graphically. There are two methods: head to tail parallelogram

Adding Vectors – Graphical Head to tail method: The tail of one vector is placed at the head of the other vector Neither the direction or length of either vector is changed A “new” vector is drawn connecting the tail of the first vector (the origin) to the head of the second vector (the end point) This new vector is the resultant vector

Adding Vectors – Graphical Any two vectors can be added graphically by using the “head to tail” method.

Adding Vectors – Graphical Let’s graphically find the sum of these two vectors using the head to tail method. B A

Adding Vectors – Graphical First, vector B must be moved so its tail (the one without the arrow point) is at the head (the one with the arrow point) of vector A. Slide vector B to that position without changing either its length (magnitude) or direction. The new position of vector B is labeled B’ in the diagram. B A B’

Adding Vectors – Graphical The resultant vector is drawn from the tail of vector A to the head of vector B and is labeled R. B A B’ R

Adding Vectors – Graphical B A B’ R The magnitude of the resultant can then be measured with a ruler. The direction could be measured with a protractor.

Adding Vectors – Graphical Simulation

Adding Vectors – Sample Problem Shown is his path. Notice all of the vectors are head to tail. A hiker walks 2 km to the North, 3 km to the West, 4 km to the South, 5 km to the East, 1 more km to the South, and finally 2 km to the West. How far did he end up from where he started? (hint: What is his resultant?) The resultant is in red. 3 km South

Adding Vectors – Sample Problem This diagram shows the same vectors being added but in a different order. Notice that the resultant is still the same.

Adding Vectors – Graphical Parallelogram method: Commonly used when you have concurrent vectors (vectors that start from the same origin) Original vectors make the adjacent sides of a parallelogram A diagonal drawn from their origin is the resultant.

Adding Vectors – Graphical The parallelogram method gives the same result as “head to tail”.

Adding Vectors – Graphical Let’s graphically find the sum of these two vectors using the parallelogram method. A B

Adding Vectors – Graphical First, vector B must be moved so its tail is at the head of vector A. Slide vector B to that position without changing either its length or direction. The new position of vector B is labeled B’ in the diagram. B A B’

Adding Vectors – Graphical Next, vector A must be moved so its tail is at the head of vector B. Slide vector A to that position without changing either its length or direction. The new position of vector A is labeled A’ in the diagram. B A B’ A’

Adding Vectors – Graphical The resultant vector is then drawn from the point where the two vectors were joined to the opposite corner of the parallelogram. This resultant is labeled R in the diagram. B A B’ A’ R

Adding Vectors – Graphical The magnitude of the resultant can then be measured with a ruler. The direction could be measured with a protractor. B A B’ A’ R

Adding Vectors by Components Any vector can also be expressed as the sum of two other vectors, called its components. usually horizontal (x) & vertical (y)

A vector is ALWAYS larger than either of its individual components. Adding Vectors by Components A vector is ALWAYS larger than either of its individual components.

Your turn! Use what you have just learned to complete the front side of today’s handout. You may NOT work in pairs or groups. If you have a question, quietly whisper to a neighbor. All questions must be ON TASK.