Vectors - It’s What’s for Dinner

Objectives Distinguish between scalar and vector quantities. Add and subtract vectors graphically. Multiply and divide vectors by scalars.

Just What IS a Vector? As you should recall, displacement, velocity, and acceleration are what we call vector quantities (there are others). To completely describe a vector quantity, we must include both magnitude and direction... 25 meters

And Why Should I Care? When you add scalar quantities, 2 + 2 ALWAYS equals 4. + =

Vectors don’t add like scalars… And Why Should I Care? And Why Should I Care? When you add scalar quantities, 2 + 2 ALWAYS equals 4. When you add vector quantities, 2 + 2 isn’t always 4... 2 N 2 N And don’t you forget it! Vectors don’t add like scalars…

A Vector is... An arrow which represents the vector quantity. The length of the vector should be proportional to the magnitude of the quantity it represents. The direction of the arrow should point in the direction of the vector quantity. Tail Tip

Vector Notation Vectors may be denoted using bold letters, or handwritten with an arrow over the top as shown below: A or

When is Vector A Equal to Vector B??? Two vectors that have the same direction and magnitude are equal to each other. A = 5 m/s B = 5 m/s

Graphical Addition of Vectors The magnitude and direction of the sum of two or more vectors can be determined by use of an accurately drawn scaled vector diagram. Using a scaled diagram, the tail-to-tip method or the parallelogram method may be employed to determine the resultant, or vector sum.

The Parallelogram Method To use this method, place vectors so that their tails are together, and construct a parallelogram. The resultant is drawn from the tails across the parallelogram… + A B A B A + B NOT A B A + B A B A + B NOT

The Tail to Tip Method A + B A + B B A A B + OR A B Using this, align vectors such that the tip of each vector touches the tail of the next vector. The resultant is then drawn from the tail of the first to the tip of the last... A B A + B + A B A B A + B OR

Practicing Tail-to-Tip This method of vector addition allows you to add multiple vectors quickly. Resultant (vector sum) Vectors are drawn to scale

Convention for Direction

And Did You Know…. … that if you multiply a vector by a scalar, you get a vector with a magnitude equal to the product of the magnitudes of the vector and scalar AND with a direction that is maintained if the scalar is positive, and 180 degrees apart if the scalar is negative: A 2A -2A

What About Subtraction? The subtraction of one vector from another requires the addition of a negative vector: A A+B A-B A B -B

More About Notation The temptation is strong to interpret as the sum of the magnitudes of the vectors but this is almost NEVER the case. A+B =5 m  7 m A=4 m B=3 m

Special Case The only time the sum of two vectors is a vector that has a magnitude equal to the sum of the magnitudes of the two vectors is when the vectors are in the same direction: A=4 m B=3 m A+B = 7 m

Adding Vectors Graphically Add vector A, which has a magnitude of 4m at an angle of 30 degrees to vector B, which has a magnitude of 3 m at an angle of 0 degrees. Draw vectors to scale, and with correct orientation, using rulers and protractors. B = 3 m A = 4 m 30º Add vectors using parallelogram or tail-to-tip method. Measure magnitude (using same scale), and reference angle.