A vector describes a movement from one point to another.
A vector quantity has both direction and magnitude.
(In contrast a scalar quantity has magnitude only - eg, the numbers 1, 2, 3, 4...)
The diagram above represents a vector. The arrow displays its direction, hence this vector can be written as , a, or .
In print, a is written in bold type. In handwriting, the vector is indicated by underlining the letter.
If we reverse the arrow it now points from B to A.
Remember that the arrow describes the direction. So, in this case, the vector is from B to A.
If we move 'backwards' along a vector, it becomes negative, so a becomes -a. Moving from B to A entails moving 3 units to the left, and 4 down.
So the three ways to write this vector are: , -a and .
If two vectors have the same magnitude and direction, then they are equal regardless of their position.
When adding vectors we follow the rule:
Look at the graph below to see the movements between PQ, QR and PR.
Vector followed by vector represents a movement from P to R.
Written out the vector addition looks like this:
Subtracting a vector is the same as adding a negative version of the vector (remember that making a vector negative means reversing its direction).
Look at the diagram and imagine going from X to Z. How would you write the path in vectors using only the vectors and ?
You could say it is vector followed by a backwards movement along .
So we can write the path from X to Z as:
Written out in numbers it looks like this:
If , and find:
A resultant vector is a vector that 'results' from adding two or more vectors together.
To travel from X to Z, it is possible to move along vector followed by . It is also possible to go directly along .
is therefore known as the resultant of and .
Write as single vectors:
2. (Did you remember the minus sign?)
Triangles ABC and XYZ are equilateral.
X is the midpoint of AB, Y is the midpoint of BC, Z is the midpoint of AC.
Express each of the following in terms of a, b and c.