Vectors

Scalar Quantities  Any measurement that consists of a single number is a scalar. 72 °F72 °F 500 milliliters500 milliliters 1.5 volts1.5 volts 6 hours, 24 minutes (consider this as 6.4 h)6 hours, 24 minutes (consider this as 6.4 h)  Most measured quantities consist of only a single magnitude.

Vector Quantities  A measurement that requires more than one value to describe it is a vector. 10 km to the northeast10 km to the northeast At 41.9° N latitude and 88.7° W longitudeAt 41.9° N latitude and 88.7° W longitude 15 pounds of force directed down15 pounds of force directed down  These quantities can be thought of as carrying the value from a starting point to a destination.  The word vector means carrier.

Vector Diagram  One representation of a vector is an arrow.  The tail shows the start of the vector.  The tip points in the direction.  The length of the arrow shows the magnitude. tail: start tip: direction length: magnitude

Vector Notation  A vector variable is represented by a small arrow over the top of the variable.  Some texts use boldface for vectors, but that can be hard to distinguish on some backgrounds. Our text uses both boldface and an arrow.  The magnitude of a vector is a scalar.  It is represented as the absolute value of the vector, or just the variable without the vector symbol.

Vectors in Equations  Vector variables can be used in equations For instance,For instance,  Vector variables are different from scalar variables They are different dimension,They are different dimension,

Graphical Addition  The two vectors can be added graphically.  The tail of the second vector is placed at the tip of the first.  The length and directions are kept the same.  The result is the total vector. Its magnitude can be measured on the graph. A = 2.0 B = 3.0 C = 4.6

Parallelogram  Force vectors act on a common object at a single point.  If two vectors are added from a common origin one can be shifted to make a parallelogram.  This is the same as putting the tail to the tip. A = 2.0 N B = 3.0 N C = 4.6 N

Commutative Property  Vectors can be shifted as long as they don’t change direction and magnitude.  Vectors can be added in reverse order and get the same result.

Parallel and Antiparallel  Vectors that point in the same direction are parallel.  Vectors that point in opposite directions are antiparallel.

Cancellation  What happens if we add two antiparallel vectors of equal magnitude?  The vector sum is a zero length vector. The vectors cancel out. next