13.01 Introduction to vectors
Whether you realise it or not, you actually use vectors very frequently in common daily activities. Vectors are quantities that differ from the quantities we have mostly used in mathematics so far called scalars.
Scalar quantities
Quantities that require only magnitude are called scalar quantities, or more simple just scalars.
- Mass
- Speed
- Time
- Temperature
- Distance
- Area
- Volume
Vector quantities
Quantities that require both magnitude and direction are called vector quantities, or more simply just vectors.
- Displacement (movement of an object from $A$A to $B$B)
- Force (and weight)
- Velocity
- Acceleration
- Momentum
Daily examples of vectors are things like walking to a certain place you have the pace that you are walking and the direction you are going. Breathing, your muscles exert a force that has both size and direction.
Vector definition
A vector is a directed line segment.
Let's look at two distinct points on the plane, $A$A and $B$B.
Then we draw in a directed line segment (a line segment with an arrow at one end), from $A$A to $B$B.
$A$A (first point) is called the initial point
$B$B (second point) is called the terminal point.
Sometimes the arrow appears at the end of the line segment like in the image above, or in the line segment like this
This is called a vector. We notate the vector using a number of different notations.
This notation indicates the initial point and the terminal point by the direction of the arrow and the order the points are written. This notation is also called vector displacement notation as it demonstrates the displacement of a point B, from A.
a The vector can be written as vector a (where the $a$a is bolded in printed text) or if you were writing by hand you would put a squiggle underneath like this, 
If the vector has a fixed point, like the origin, $O$O, then it is called a position vector. Position vectors can also be represented using matrix notation. For example,
can be represented 
and
can be represented 
Note that the first element in the column matrix indicates the $x$x movement (right is positive and left is negative) and the second element indicates the change in $y$y, (up positive and down negative).
This applet will give you some practice at making vectors, (uses column vector notation)
Negative of a vector
As with most things we have learnt about in mathematics, a negative sign indicates the opposite direction. A negative vector relates to a vector that has the same magnitude (size), but goes in the opposite direction.
This applet will give you some practice at making negative vectors.
Zero vectors
A zero vector has a zero magnitude and any direction, effectively it is a point on the plane.
Worked examples
example 1
Which of the following are vector quantities?
a force of 8 N acting horizontally right
Aa displacement of 4 m along the line joining A and B
Ba mass of 1 kg
Ca time of 1 seconds
D
Remember that a vector quantity has to have both magnitude (size) AND direction.
All these options have magnitude.
Only $(A)$(A) and $(B)$(B) have direction.
$(A)$(A) is in the direction given by "horizontally right", and $(B)$(B) is in the direction "along the line joining A and B".
example 2
Write the vector represented on the plane as a column vector.
$\editable{}$ $\editable{}$
The first component in the column vector is the $x$x-component of the vector. This vector travels $4$4 units in the positive $x$x-direction.
The second component in the column vector is the $y$y-component. This vector travels $4$4 units in the positive $y$y-direction.
So the column vector required is 
example 3
| Plot the vector |
|
. Use the origin as the starting point for the vector. |
- Loading Graph...
Component form
It is often convenient to use a special pair of vectors as the basis for the vectors in the plane. These are two vectors, called $\mathbf{i}$i and $\mathbf{j}$j, that are at right-angles to one another and that have unit length. In terms of the Cartesian coordinate system, the vector $\mathbf{i}$i points in the direction usually designated the $x$x-axis, and the vector $\mathbf{j}$j points in the direction usually designated the $y$y-axis.
This convention makes it very easy to switch between a representation of a vector as a coordinate pair and an equivalent representation as a linear combination of basis vectors (which we will look at in more detail later).
Worked examples
Example 4
The coordinate pair $(4,3)$(4,3) represents the same vector as the sum $4\mathbf{i}+3\mathbf{j}$4i+3j.
Care is needed if the vector under consideration does not have its tail at the origin of the coordinate system. Remember that an arrow translated so that its length and direction do not change, is considered to be the same vector.
Example 5
The vector represented by the green arrow is clearly the same vector as in the previous diagram. You should look at this diagram carefully to make sure that you understand why the two vectors are the same.
Practice questions
Question 1
Rewrite the vector $\left(0,9\right)$(0,9) in terms of the unit vectors $i$i and $j$j.
Question 2
Rewrite the vector $-8i$−8i as an ordered pair.
Question 3
Write the vector on the graph in terms of the unit vectors $i$i and $j$j.
Magnitude of a vector
The magnitude of a vector is the size, or length, of a vector. The magnitude of a vector is indicated using absolute value notation. So magnitude of vector $\overrightarrow{AB}$›‹AB is $\left|\overrightarrow{AB}\right|$|›‹AB|.
Magnitude from initial and terminal points
The magnitude of a vector, given the coordinates of the initial and terminal points, can be calculated using the distance formula:
The magnitude of the vector $\overrightarrow{AB}$›‹AB with initial point $A\left(x_1,y_1\right)$A(x1,y1) and terminal point $B\left(x_2,y_2\right)$B(x2,y2) is given by:
| $\left|\overrightarrow{AB}\right|$|›‹AB| | $=$= | $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$√(x2−x1)2+(y2−y1)2 |
Practice question
Question 4
Consider the vector defined by the directed line segment from $\left(-5,3\right)$(−5,3) to $\left(1,-5\right)$(1,−5).
Plot the vector.
Loading Graph...Find the magnitude of the vector.
Magnitude using column vectors and ordered pairs
The magnitude of vectors in column vector notation or as ordered pairs is given by:
For $\mathbf{u}=\left(a,b\right)=\binom{a}{b}$u=(a,b)=(ab):
| $\left|\mathbf{u}\right|$|u| | $=$= | $\sqrt{a^2+b^2}$√a2+b2 |
Practice question
Question 5
Find the magnitude of the vector
| $-7$−7 | ||
| $0$0 |
Magnitude using vectors in component form
The magnitude of vectors in component form can be found using:
For $\mathbf{u}=a\mathbf{i}+b\mathbf{j}$u=ai+bj:
| $\left|\mathbf{u}\right|$|u| | $=$= | $\sqrt{a^2+b^2}$√a2+b2 |
Practice question
Question 6
Find the magnitude of the vector $12i+16j$12i+16j.
Direction of a vector
The direction of a vector is defined as the angle a vector makes with the positive direction of the horizontal coordinate axis. The angle is measured anti-clockwise from the horizontal axis to the vector.
Direction from initial and terminal points
The direction of a vector, given the coordinates of the initial and terminal points, can be calculated using trigonometry:
The direction of the vector $\overrightarrow{AB}$›‹AB with initial point $A\left(x_1,y_1\right)$A(x1,y1) and terminal point $B\left(x_2,y_2\right)$B(x2,y2) is given by:
| $\tan\theta$tanθ | $=$= | $$ |
Direction using column vectors and ordered pairs
The direction of vectors in column vector notation or as ordered pairs is given by:
For $\mathbf{u}=\left(a,b\right)=\binom{a}{b}$u=(a,b)=(ab):
| $\tan\theta$tanθ | $=$= | $\frac{b}{a}$ba |
Practice question
Question 7
Consider the vector $\left(2\sqrt{3},2\right)$(2√3,2).
Find the magnitude of the vector.
Find the direction of the vector.
Give your answer in degrees.
Direction using vectors in component form
The magnitude of vectors in component form can be found using:
For $\mathbf{u}=a\mathbf{i}+b\mathbf{j}$u=ai+bj:
| $\tan\theta$tanθ | $=$= | $\frac{b}{a}$ba |
Practice question
Question 8
Let $U=5i-3j$U=5i−3j be a vector.
Find the exact magnitude of the vector $U$U:
Determine the direction angle $\theta$θ of vector $U$U in degrees, where $0^\circ\le\theta<360^\circ$0°≤θ<360°. Give you answer correct to two decimal places.