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IB DP Maths: AI HL復(fù)習(xí)筆記3.8.1 Vector Equations of Lines

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月20日下午5:44

The formula for finding the?vector equation?of a line is
- This is?given in the formula booklet
- This equation can be used for vectors in both 2- and 3- dimensions
  - In 2D the gradient can be found from the direction vector
  - In 3D a numerical value for the direction cannot be found, it is given as a vector
As?a?could be the position vector of?any?point on the line and?b?could be?any scalar multiple?of the direction vector there are infinite vector equations for a single line
Given any two points on a line with position vectors?a?and?b?the?displacement?vector can be written as?b?-?a
- So the formula?r = a?+?λ(b?-?a)?can be used to find the vector equation of the line
- This is?not given in the formula booklet

Remember that the vector equation of a line can take many different forms
- This means that the answer you derive might look different from the answer in a mark scheme
You can choose whether to write your vector equations of lines using unit vectors or as column vectors
- Use the form that you prefer, however column vectors is generally easier to work with

a)Find a vector equation of a straight line through the points with position vectors?a?= 4i?– 5k?and?b?= 3i?- 3k

b)Determine whether the point C with coordinate (2, 0, -1) lies on this line.

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By considering the three separate components of a vector in the?x, y?and?z?directions it is possible to write the?vector equation?of a line as?three separate equations
- These are?given in the formula booklet

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The angle between two lines is equal to the angle between their?direction vectors
- It can be found using the?scalar product?of their direction vectors
If you are given the equations of the lines in a different form or two points on a line you will need to find their direction vectors first
To find the angle ABC the vectors BA and BC would be used, both starting from the point B
The intersection of two lines will always create?two angles, an acute one and an obtuse one
- A?positive scalar product?will result in the?acute angle?and a?negative scalar product?will result in the?obtuse angle
  - Using the?absolute value?of the scalar product will?always result in the acute angle

In your exam read the question carefully to see if you need to find the acute or obtuse angle
- When revising, get into the practice of double checking at the end of a question whether your angle is acute or obtuse and whether this fits the question