Home » 國(guó)際課程 » IB課程 » Details

IB DP Maths: AA HL復(fù)習(xí)筆記3.9.5 Geometric Proof with Vectors

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月12日下午4:52

Geometrical Proof with Vectors

How can vectors be used to prove geometrical properties?

If two vectors can be shown to be?parallel?then this can be used to prove parallel lines
- If two vectors are?scalar multiples?of each other then they are?parallel
- To prove that two vectors are parallel simply show that one is a scalar multiple of the other
If two vectors can be shown to be?perpendicular?then this can be used to prove perpendicular lines
- If the?scalar product?is zero then the two vectors are?perpendicular
If two vectors can be shown to have equal?magnitude?then this can be used to prove two lines are the?same?length
To prove a 2D shape is a?parallelogram?vectors can be used to
- Show that there are two pairs of?parallel sides
- Show that the?opposite sides?are of?equal length
  - The vectors opposite each other with be?equal
- If the angle between two of the vectors is shown to be 90° then the parallelogram is a?rectangle
To prove a 2D shape is a?rhombus?vectors can be used to
- Show that there are two pairs of?parallel sides
  - The vectors opposite each other with be?equal
- Show that?all four sides?are of?equal length
- If the angle between two of the vectors is shown to be 90° then the rhombus is a?square

How are vectors used to follow paths through a diagram?

How can vectors be used to find the midpoint of two vectors?

Exam Tip

Think of vectors like a journey from one place to another
- You may have to take a detour e.g. A to B might be A to O then O to B
Diagrams can help, if there isn’t one, draw one
- If a diagram has been given begin by labelling all known quantities and vectors

Worked Example

Use vectors to prove that the points?A, B, C?and?D?with position vectors?a?= (3i?+ 5j?– 4k),?b?= (8i?- 7j?- 5k),?c?= (3i?- 2j?+ 4k) and?d?= (5k?– 2i) are the vertices of a parallelogram.