Let $\,k\,$ be a real number (a scalar).
Then, $\,k\vec v\,$ is a vector.
That is, a scalar times a vector produces a vector.
In other words, a real number times a vector produces a vector.
The image below illustrates the relationship between an original vector $\,\vec v\,$ and various scaled versions:
Multiplying a vector by a positive number $\,k\,$:
- doesn't change the direction of the vector
-
changes the length by a factor of $\,k\,$
For example:
- if $\,k = 2\,,$ the length is doubled
- if $\,k=\frac 12\,,$ the length is halved
|
Multiplying a vector by a negative number $\,k\,$:
- produces a vector pointing in the opposite direction
-
changes the length by a factor of $\,|k|\,$
For example:
- if $\,k = -2\,,$ the length is doubled,
and the vector points in the opposite direction
- if $\,k=-\frac 12\,,$ the length is halved,
and the vector points in the opposite direction
|
Multiplying any vector by the real number $\,0\,$ produces the zero vector.
For all vectors $\,\vec v\,,$
$$\cssId{s27}{-\vec v\, = -1\cdot\vec v}$$
is the opposite of $\,\vec v\,.$
|
Finding the Size of a Scaled Vector
To find the size of a scaled vector, you multiply together two numbers:
- the absolute value (size) of the scaling constant
- the size of the original vector
Precisely, we have:
For all real numbers $\,k\,,$ and for all vectors $\,\vec v\,,$
$$
\cssId{s35}{\| k \vec v\| = |k|\,\cdot\,\|\vec v\|}
$$
Keep in mind:
- Whenever you see $\,|\cdot|\,,$ there must be a real number inside.
- Whenever you see $\,\|\cdot\|\,,$ there must be a vector inside.
For example:
$\|3\vec v\|$ |
$=$ |
$|3|\cdot\|\vec v\|$ |
$=$ |
$3\|\vec v\|$ |
| | and | | |
$\|-3\vec v\|$ |
$=$ |
$|-3|\cdot\|\vec v\|$ |
$=$ |
$3\|\vec v\|$ |
So,
$\|-3\vec v\| = \|3\vec v\|\,.$
A vector and its opposite have the same length.
Adding Vectors (‘Head-to-Tail Addition’)
Adding the arrow representations of vectors is done using the ‘head-of-first to tail-of-second’ rule.
This is usually abbreviated as ‘head-to-tail addition’.
Here's how to add $\,\vec u\,$ to $\,\vec v\,$:
- draw an arrow representing $\,\vec u\,$
- to the head of $\,\vec u\,,$ attach the tail of vector $\,\vec v\,$
- the sum $\,\vec u+\vec v\,$ goes from the tail of $\,\vec u\,$ to the head of $\,\vec v\,$
It sounds complicated when written out.
The diagram below shows how simple it really is:
|
Vector Addition is Commutative:
-
the configuration to find $\vec u + \vec v\,$ is shown in red:
head of $\,\vec u\,$ is attached to tail of $\,\vec v\,$
-
the configuration to find $\vec v + \vec u\,$ is shown in blue:
head of $\,\vec v\,$ is attached to tail of $\,\vec u\,$
-
in both cases, going from the tail of the first to the head of the second
gives the same vector (shown in black)
-
So, $\,\vec u + \vec v = \vec v + \vec u\,.$
This always works!
Vector addition is commutative.
|
|
Vector Addition is Associative:
-
the configuration to find $(\vec u + \vec v) + \vec w\,$ is shown in red:
head of $\,\vec u + \vec v\,$ is attached to tail of $\,\vec w\,$
-
the configuration to find $\vec u + (\vec v + \vec w)\,$ is shown in blue:
head of $\,\vec u\,$ is attached to tail of $\,\vec v + \vec w\,$
-
in both cases, going from
the tail of the first to the head of the last
gives the same vector (shown in green)
-
So, $\,(\vec u + \vec v) + \vec w = \vec u + (\vec v + \vec w)\,.$
This always works!
Vector addition is associative.
Therefore, we can write $\,\vec u + \vec v + \vec w\,$ (no parentheses) without ambiguity.
-
To add any number of vectors, do head-to-tail addition in any order.
The vector from the tail of the first to the head of the last is the vector sum.
|
Subtracting Vectors
To subtract a vector, just add its opposite:
$$\,\cssId{s86}{\vec u - \vec v := \vec u + (-\vec v)}\,$$
(Remember that ‘$\,:=\,$’ means ‘equals, by definition’).
Other Operations with Vectors
-
There are different types of vectors.
The vectors we're talking about here are two-dimensional vectors—vectors in a plane.
There are also three-dimensional vectors—vectors in space—and lots more.
When you add vectors, they have to be vectors of the same type.
-
Depending on what type of vectors you're working with, there may be other
operations defined.
For example, there is a ‘dot product’ and a
‘cross product’ that you'll likely come across if you study Calculus.