Vectors

1
Vectors
2
Scalar Quantities

• Any measurement that consists of a single number
  is a scalar.
• 72 F
• 500 milliliters
• 1.5 volts
• 6 hours, 24 minutes (consider this as 6.4 h)
• Most measured quantities consist of only a single
  magnitude.

3
Vector Quantities

• A measurement that requires more than one value
  to describe it is a vector.
• 10 km to the northeast
• At 41.9 N latitude and 88.7 W longitude
• 15 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.

4
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.

tip direction
length magnitude
tail start
5
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.

6
Vectors in Equations

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

7
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.

C 4.6
B 3.0
A 2.0
8
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.

C 4.6 N
B 3.0 N
A 2.0 N
9
Commutative Property

• Vectors can be shifted as long as they dont
  change direction and magnitude.
• Vectors can be added in reverse order and get the
  same result.

10
Parallel and Antiparallel

• Vectors that point in the same direction are
  parallel.

• Vectors that point in opposite directions are
  antiparallel.

11
Cancellation

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

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