# Edexcel A Level Further Maths: Core Pure:复习笔记6.1.4 Shortest Distances - Lines

### Shortest Distance between a Point & a Line

#### How do I find the shortest distance from a point to a line?

• The shortest distance from any point to a line will always be the perpendicular distance
• Given a line l  with equation   and a point P not on l
• The scalar product of the direction vector, b, and the vector in the direction of the shortest distance will be zero
• The shortest distance can be found using the following steps:
• STEP 1: Let the vector equation of the line be r and the point not on the line be P, then the point on the line closest to P will be the point F
• The point F is sometimes called the foot of the perpendicular
• STEP 2: Sketch a diagram showing the line l and the points P and F
• The vector  will be perpendicular to the line l
• STEP 3: Use the equation of the line to find the position vector of the point F  in terms of λ
• STEP 4: Use this to find the displacement vector  in terms of λ
• STEP 5: The scalar product of the direction vector of the line l and the displacement vector  will be zero
• Form an equation  and solve to find λ
• STEP 6: Substitute λ into  and find the magnitude
• The shortest distance from the point to the line will be the magnitude of
• Note that the shortest distance between the point and the line is sometimes referred to as the length of the perpendicular

#### Exam Tip

• Column vectors can be easier and clearer to work with when dealing with scalar products.

#### Worked Example

Point A  has coordinates (1, 2, 0) and the line  has equation .

Find the shortest distance from A to the line .

### Shortest Distance between two Lines

#### How do we find the shortest distance between two parallel lines?

• Two parallel lines will never intersect
• The shortest distance between two parallel lines will be the perpendicular distance between them
• Given a line with equation and a line with equation  then the shortest distance between them can be found using the following steps:
• Remember the direction vectors  and  are scalar multiples of each other and so either can be used here
• STEP 1: Find the vector between and a general coordinate from  in terms of μ
• STEP 2: Set the scalar product of the vector found in STEP 1 and the direction vector equal to zero
• STEP 3: Form and solve an equation to find the value of μ
• STEP 4: Substitute the value of μ  back into the equation for to find the coordinate on  closest to
• STEP 5: Find the distance between  and the coordinate found in STEP 4

#### How do we find the shortest distance from a given point on a line to another line?

• The shortest distance from any point on a line to another line will be the perpendicular distance from the point to the line
• If the angle between the two lines is known or can be found then right-angled trigonometry can be used to find the perpendicular distance
• Alternatively, the equation of the line can be used to find a general coordinate and the steps above can be followed to find the shortest distance

#### How do we find the shortest distance between two skew lines?

• Two skew lines are not parallel but will never intersect
• The shortest distance between two skew lines will be perpendicular to both of the lines
• To find the shortest distance between two skew lines with equations and  ,
• STEP 1: Find position vectors for the points on each line that form the shortest distance
• Point P has position vector
• Point Q has position vector
• STEP 2: Find the displacement vector between P and Q
• STEP 3: Form two equations by using the fact that the scalar product of the displacement vector and the direction vector of each line should equal zero
• STEP 4: Solve the two equations simultaneously to find the values of λ and μ
• STEP 5: Substitute the values of λ and μ into the displacement vector and take the magnitude
• Shortest distance =

#### Exam Tip

• Exam questions will often ask for the shortest, or minimum, distance within vector questions
• If you’re unsure start by sketching a quick diagram
• Sometimes calculus can be used, however usually vector methods are required

#### Worked Example

Consider the skew lines and  as defined by:

Find the minimum distance between the two lines.