Geometric problem solving - Higher
Geometric problems can be solved using the rules for adding and subtracting A vector describes a movement from one point to another. A vector quantity has magnitude (size) and direction. and multiplying vectors by a A quantity that requires only a size, for example, distance travelled is 20 m..
Example
OABC is a parallelogram. \(\overrightarrow{OA}\) is represented by the vector \(\mathbf{a}\) and \(\overrightarrow{OC}\) is represented by the vector \(\mathbf{c}\).
M is the mid-point of BC and N is the point on OB such that ON:NB = 2:1.
- Find expressions for these vectors, giving the answers in the simplest form: \(\overrightarrow{ON}\), \(\overrightarrow{OM}\), \(\overrightarrow{AN}\) and \(\overrightarrow{AM}\).
- Show that points A, N and M lie on a straight line.
Part 1
Point N is \(\frac{2}{3}\) of the distance from O to B.
\(\overrightarrow{ON} = \frac{2}{3} \overrightarrow{OB}\)
\(\overrightarrow{OB} = \mathbf{a} + \mathbf{c}\)
\(\overrightarrow{ON} = \frac{2}{3}(\mathbf{a} + \mathbf{c})\)
Point M is half the distance from C to B.
\(\overrightarrow{CM} = \frac{1}{2} \overrightarrow{CB}\)
\(\overrightarrow{CM} = \frac{1}{2} \mathbf{a}\)
\(\overrightarrow{OM} = \overrightarrow{OC} + \overrightarrow{CM}\)
\(\overrightarrow{OM} = \mathbf{c} + \frac{1}{2} \mathbf{a}\)
\(\overrightarrow{AN} = \overrightarrow{AO} + \overrightarrow{ON}\)
\(\overrightarrow{AN} = \mathbf{-a} + \frac{2}{3} (\mathbf{a} + \mathbf{c}) = -\frac{1}{3} \mathbf{a} + \frac{2}{3} \mathbf{c}\)
\(\overrightarrow{AM} = \overrightarrow{AO} + \overrightarrow{OM}\)
\(\overrightarrow{AM} = \mathbf{-a} + \frac{1}{2} \mathbf{a} + \mathbf{c} = -\frac{1}{2} \mathbf{a} + \mathbf{c}\)
Part 2
To prove that points A, N and M lie on a straight line:
Show \(\overrightarrow{AM}\) is a multiple of \(\overrightarrow{AN}\).
\(\frac{3}{2} \overrightarrow{AN} = \frac{3}{2} (-\frac{1}{3} \mathbf{a} + \frac{2}{3} \mathbf{c}) = -\frac{1}{2} \mathbf{a} + \mathbf{c} = \overrightarrow{AM}\)
Therefore \(\overrightarrow{AM}\) and \(\overrightarrow{AN}\) are Straight lines are parallel if they are always the same distance apart. Parallel lines never meet, no matter how far they are extended.. They also share a common point A so they lie on the same straight line.
Points which lie on the same straight line are called collinear.