Introduction to Vectors Unit 2 Presentation 1

What is a vector? Vector: A quantity that contains both a magnitude and a direction.  Represented by a small arrow sign above the variable: Scalar: A quantity that contains only a magnitude.  Represented WITHOUT any sign above variable.

More Vectors Vectors are equal ONLY IF they have the same magnitude AND the same direction. x y In the example Coordinate Plane on the right, consider the following 4 vectors: The orange and green vectors are equal because they have the same magnitude (length) and the same direction. The pink vector is in the same direction as the orange and green vectors but is only half the magnitude The blue vector has the same magnitude as the orange and green vectors, but is in the opposite direction. Hence, it is the opposite vector from the orange and green vectors.

Opposite Vectors Vectors are defined to be opposite of each other when the sum of the two equals zero. Opposite vectors have the same magnitude, but are pointed in opposite directions.

Adding Vectors Geometrically x y x y First, put the tail end (not the pointed end) of the first vector ( in this case) at the origin Second, put the tail end of the second vector ( in this case) at the head end of the first vector. Lastly, draw a vector from the origin to the head end of the final vector. This “vector” that you have drawn yourself (in green above) is the vector sum and if your answer. +

Adding Vectors Geometrically Can you add more than one vector geometrically? YES! Just keep adding the tail of your next vector to the head of the last vector, remembering to start with the first vector at the origin. x y A x y B C D A+B+C+D y A B C D

Subtracting Vectors Geometrically To subtract two vectors, say A-B, simply find the opposite of B and add it: A+(-B): x y A B x y A -B A-B

Multiplying a Vector by a Scalar To multiply a vector by a scalar, simply multiply the magnitude by the scalar  Don’t change the direction in any way. x y A 3A 0.33A -2A

Breaking a Vector into Components Two ways to mathematically represent a vector:  Magnitude + Direction (think Polar Coordinates, (r,   30 15°  °  Component Form (think Rectangular Coordinates, (x, y)  X and Y components, both magnitudes  Represented either by or x Î (i-hat), y ĵ (j-hat)  For example:  (or also written as 15 m Î, 30 m ĵ  (or also written as -20 m/s Î, 0.5 m/s ĵ)  Note that units are written on BOTH numbers

Vectors into Components Graphically x y Vector A = 30 60° X-component of Vector A X=30 m * cos(60°) X=15 m Y-component of Vector A Y = 30 m * sin(60°) Y=25.98 m Hence, Vector A can also be written as 60°

Adding Vectors Algebraically Can NOT be done in Magnitude + Direction Form Easily done in Component Form For example:  Add with  Simply add the respective components together   Answer = Remember, you can convert between the two forms!

Multiplying a Vector by a Scalar To multiply a vector by a scalar, simply multiply each component by the scalar:  3 * =  Answer = To subtract two vectors, simply add the opposite of the second vector:  -  + = As we did in the geometric interpretation, the same rules hold true for more than 2 vectors.

A Complex Vector Example Bob walks 1500 m at a bearing of 30°, then turns and walks 2000 m at a bearing of 70°. Bob then wants to walk home in one straight line. How far does Bob have to walk, and in what direction? First, draw a picture. Find the sum of these two vectors, and the answer will be the opposite of this sum vector. y 70° 30° Next, break both vectors down into their components: A x = 1500 km * cos (30°) = 1299 m A y = 1500 km * sin (30°) = 750 m B x = 2000 km * cos (70°) = 684 m B y = 2000 km * sin (70°) = 1879 m A B A + B

A Complex Vector Example (cntd) Now, add the components together: Answer = Now, convert this vector back into Magnitude + Direction Form r = SQRT((1983) 2 + (2629) 2 ) = 3293 m  = tan -1 (2629 / 1983) = 53° But, consider that Bob wants to walk BACK to his car, so add 180° to the direction = 233° FINAL ANSWER: Bob must walk 3293 m at 233°.