Geometric vectors
A vector describes a movement from one point to another.
Vector notation
A vector quantity has both direction and magnitudeThe magnitude tells us the size of the vector..
(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 \(\overrightarrow {AB}\), a, or \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\).
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: \(\overrightarrow {BA}\), -a and \(\begin{pmatrix} -3 \\ -4 \end{pmatrix}\).
Equal vectors
If two vectors have the same magnitude and direction, then they are equal regardless of their position.
Adding vectors
When adding vectors we follow the rule:
\(\left( \begin{array}{l} a\\ b \end{array} \right) + \left( \begin{array}{l} c\\ d \end{array} \right) = \left( \begin{array}{l} a + c\\ b + d \end{array} \right)\)
Look at the graph below to see the movements between PQ, QR and PR.
Vector \(\overrightarrow {PQ}\) followed by vector \(\overrightarrow {QR}\) represents a movement from P to R.
\(\overrightarrow {PQ} + \overrightarrow {QR} = \overrightarrow {PR}\)
Written out the vector addition looks like this:
\(\left( \begin{array}{l}2\\5\end{array} \right) + \left( \begin{array}{l}\,\,\,\,\,4\\- 3\end{array} \right) = \left( \begin{array}{l}6\\2\end{array} \right)\)
Subtracting vectors
Subtracting a vector is the same as adding a negative version of the vector (remember that making a vector negative means reversing its direction).
\(\left( \begin{array}{l} a\\ b \end{array} \right) - \left( \begin{array}{l} c\\ d \end{array} \right) = \left( \begin{array}{l} a - c\\ b - d \end{array} \right)\)
Look at the diagram and imagine going from X to Z. How would you write the path in vectors using only the vectors \(\overrightarrow {XY}\) and \(\overrightarrow {ZY}\)?
You could say it is vector \(\overrightarrow {XY}\) followed by a backwards movement along \(\overrightarrow {ZY}\).
So we can write the path from X to Z as:
\(\overrightarrow {XY} - \overrightarrow {ZY} = \overrightarrow {XZ}\)
Written out in numbers it looks like this:
\(\left( \begin{array}{l} 4\\ 2 \end{array} \right) - \left( \begin{array}{l} 1\\ 2 \end{array} \right) = \left( \begin{array}{l} 3\\ 0 \end{array} \right)\)
Question
If \(x = \left( \begin{array}{l} 1\\ 3 \end{array} \right)\), \(y = \left( \begin{array}{l} - 2\\ 4 \end{array} \right)\) and \(z = \left( \begin{array}{l} - 1\\ - 2 \end{array} \right)\) find:
- \(- y\)
- \(x - y\)
- \(2x + 3z\)
- \(\left( \begin{array}{l}\,\,\,\,\,2\\- 4\end{array} \right)\) Did you remember to change the signs?
- \(\left( \begin{array}{l}\,1\\3\end{array} \right) - \left( \begin{array}{l}- 2\\\,\,\,\,4\end{array} \right) = \left( \begin{array}{l}\,1 - - 2\\3 - \,\,\,\,4\end{array} \right) = \left( \begin{array}{l}\,\,\,\,\,3\\- 1\end{array} \right)\)
- \(\left( \begin{array}{l}\,1\\3\end{array} \right) + 3\left( \begin{array}{l}- 1\\- 2\end{array} \right) = \left( \begin{array}{l}2\\6\end{array} \right) + \left( \begin{array}{l}- 3\\- 6\end{array} \right) = \left( \begin{array}{l}- 1\\\,\,\,\,0\end{array}\right)\)
Resultant vectors
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 \(\overrightarrow {XY}\) followed by \(\overrightarrow {YZ}\). It is also possible to go directly along \(\overrightarrow {XZ}\).
\(\overrightarrow {XZ}\) is therefore known as the resultant of \(\overrightarrow {XY}\) and \(\overrightarrow {YZ}\) .
Question
Write as single vectors:
1. \(f + g\)
2. \(a + b\)
3. \(e - b - a\)
1. \(e\)
2.\(- c\) (Did you remember the minus sign?)
3. \(- d\)
Question
Triangles ABC and XYZ are equilateral.
X is the midpoint of AB, Y is the midpoint of BC, Z is the midpoint of AC.
\(\overrightarrow {AX} = a\), \(\overrightarrow {XZ} = b\), \(\overrightarrow {AZ} = c\)
Express each of the following in terms of a, b and c.
- \(\overrightarrow {XY}\)
- \(\overrightarrow {YZ}\)
- \(\overrightarrow {XC}\)
- \(\overrightarrow {BZ}\)
- \(\overrightarrow {AC}\)
- c
- - a Remember that \(\overrightarrow {YZ}\) is parallelStraight lines are parallel if they are always the same distance apart. Parallel lines never meet, no matter how far they are extended. to \(\overrightarrow {AX}\) and of the same length, but the direction is different.
- b + c (It is also possible to move from X to A and then on to C. This would give the answer - a + 2c. How many other answers can you think of?)
- b - a or 2b - c or - 2a + c
- 2c