IB DP Maths: AI HL復習筆記3.8.1 Vector Equations of Lines
Equation of a Line in Vector Form
How do I find the vector equation of a line?
- 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
How do I determine whether a point lies on a line?
Exam Tip
- 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
Worked Example
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.
Equation of a Line in Parametric Form
How do I find the vector equation of a line in parametric form?
- 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
Worked Example
Angle Between Two Lines
How do we find the angle between two lines?
- 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
- A?positive scalar product?will result in the?acute angle?and a?negative scalar product?will result in the?obtuse angle
Exam Tip
- 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
Worked Example
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