IB DP Maths: AI HL复习笔记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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