Introduction to Vectors

PRACTICE (online exercises and printable worksheets)

Vectors versus Scalars


Some quantities can be described by a single number;
such quantities are called scalar quantities.
Scalar quantities are completely described by their size.
Size is also called magnitude.

If you get complete information by asking ‘How much?’
then you're dealing with a scalar quantity.
The answer may also include a unit,
but there will be a single number.

Area, time, and volume are examples of scalar quantities:

  • How much area? Answer: $\,2\,$ square feet
  • How much time? Answer: $\,3\,$ seconds
  • How much volume? Answer: $\,4\,$ cubic inches


A quantity that requires both size and direction to fully describe it is called a vector quantity or—more simply—a vector.

For vector quantities, two questions must be answered to get complete information:

  • a question that gives size, like:
    How much?   How fast?   How strong?
  • a second question: What direction?
Velocity and force are examples of vector quantities:
  • Velocity:
    If an object moves, two things are of interest:
    How fast is it going? (size)
    In what direction?
  • Force:
    If you (say) push an object, two things are of interest:
    How hard are you pushing? (size)
    In what direction are you pushing?

Representation of Vectors

Size (Magnitude) of a Vector


The size (magnitude) of a vector
is the length of the arrow representing it.

Size is notated by enclosing the vector inside double vertical bars:
for example,
$\text{size of } \overrightarrow{AB} = \|\overrightarrow{AB}\|\,$
$\text{size of } \vec v = \|\vec v\|$

The vectors shown above are all different,
because they all have different directions.
However, they all have the same size, which is: $\sqrt{2^2 + 5^2} = \sqrt{29}$
  • The size of a vector is the length of the arrow representing it.
  • In common usage, the words size, magnitude, and length are all used interchangeably in this context.
    So, these are all the same:
    • the size of a vector
    • the magnitude of a vector
    • the length of a vector
  • The size of a vector is notated by putting the vector inside vertical bars. For example:
    The size of vector $\,\overrightarrow{AB}\,$ is notated by $\,\|\overrightarrow{AB}\|\,$.
    The size of vector $\,\vec v\,$ is notated by $\,\|\vec v\|\,$.
  • Note:
    • The size of a real number $\,x\,$ is denoted using absolute value: $\,|x|\,$
    • The size of a vector $\,\vec v\,$ is denoted by $\,\|\vec v\|\,$.
    • Some people use ‘$\,|\cdot|\,$’ for both real numbers and vectors. This is called ‘operator overloading’ in computer science. For beginning math students, it's easiest (and safer) to use different notation.
  • Since size gives the length of an arrow,
    it is a nonnegative quantity:

    For all vectors $\,\vec v\,$, $\,\|\vec v\| \ge 0\,$.
  • The zero vector is the unique vector of length zero.
    If a vector has length zero, then it is the zero vector.
    That is:
    $\,\vec v\,$ is the zero vector     if and only if     $\,\|\vec v\| = 0$

Equality of Vectors

  • By definition, vectors are equal when they have the same size and the same direction.
  • All the vectors shown at left are equal, since they all have the same size and direction.
  • In particular, to decide if two vectors are equal, it doesn't matter where they start. All that matters is if they have the same length and the same direction.
  • The consequence is that vectors are portable.
    They can be moved to wherever they're needed!
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Working with the Arrow Representation for Vectors

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
1 2 3 4 5 6 7 8 9 10 11

(MAX is 11; there are 11 different problem types.)
Want textboxes to type in your answers? Check here: