3.11.4 Shortest Distances with Planes

Shortest Distance Between a Line and a Plane

How do I find the shortest distance between a given point on a line and a plane?

The shortest distance from any point on a line to a plane will always be the perpendicular distance from the point to the plane
Given a point, P, on the line $l$ with equation $r = a + λ b$ and a plane $Π$ with equation $r \cdot n = d$
- STEP 1: Find the vector equation of the line perpendicular to the plane that goes through the point, P, on $l$
  - This will have the position vector of the point, P, and the direction vector n
- STEP 2: Find the coordinates of the point of intersection of this new line with $Π$ by substituting the equation of the line into the equation of the plane
- STEP 3: Find the distance between the given point on the line and the point of intersection
  - This will be the shortest distance from the plane to the point
A question may provide the acute angle between the line and the plane
- Use right-angled trigonometry to find the perpendicular distance between the point on the line and the plane
  - Drawing a clear diagram will help

ZAFnkWOW_3-11-4-ib-hl-aa-shortest-dist-two-planes-diagram-1

How do I find the shortest distance between a plane and a line parallel to the plane?

The shortest distance between a line and a plane that are parallel to each other will be the perpendicular distance from the line to the plane
Given a line $l_{1}$ with equation $r = a + λ b$ and a plane $Π$ parallel to $l_{1}$ with equation $r \cdot n = d$
- Where n is the normal vector to the plane

STEP 1: Find the equation of the line $l_{2}$ perpendicular to $l_{1}$ and $Π$ going through the point a in the form $r = a + μ n$
STEP 2: Find the point of intersection of the line $l_{2}$ and $Π$
STEP 3: Find the distance between the point of intersection and the point,

3-11-4-ib-hl-aa-shortest-dist-two-planes-diagram-2-1

Exam Tip

Vector planes questions can be tricky to visualise, read the question carefully and sketch a very simple diagram to help you get started

Worked Example

The plane $Π$ has equation $r \cdot (\begin{matrix} 2 \\ - 1 \\ 1 \end{matrix}) = 6$ .

The line $L$ has equation $r = (\begin{matrix} 2 \\ 3 \\ 1 \end{matrix}) + s (\begin{matrix} 1 \\ - 2 \\ 4 \end{matrix})$ .

The point $P (- 2, 11, - 15)$ lies on the line $L$ .

Find the shortest distance between the point P and the plane $Π$ .

3-11-4-ib-hl-aa-shortest-dist-two-planes-we-1

Shortest Distance Between Two Planes

How do I find the shortest distance between two parallel planes?

Two parallel planes will never intersect
The shortest distance between two parallel planes will be the perpendicular distance between them
Given a plane $Π_{1}$ with equation $r \cdot n = d$ and a plane $Π_{2}$ with equation $r = a + λ b + μ c$ then the shortest distance between them can be found

STEP 1: The equation of the line perpendicular to both planes and through the point a can be written in the form r = a + sn
STEP 2: Substitute the equation of the line into $r \cdot n = d$ to find the coordinates of the point where the line meets $Π_{1}$
STEP 3: Find the distance between the two points of intersection of the line with the two planes

How do I find the shortest distance from a given point on a plane to another plane?

The shortest distance from any point, P on a plane, $Π_{1}$ , to another plane, $Π_{2}$ will be the perpendicular distance from the point to $Π_{2}$

STEP 1: Use the given coordinates of the point P on $Π_{1}$ and the normal to the plane $Π_{2}$ to find the vector equation of the line through P that is perpendicular to $Π_{1}$
STEP 2: Find the point of intersection of this line with the plane $Π_{2}$
STEP 3: Find the distance between the two points of intersection

Exam Tip

There are a lot of steps when answering these questions so set your methods out clearly in the exam

Worked Example

Consider the parallel planes defined by the equations:

$Π_{1} : r \cdot (\begin{matrix} 3 \\ - 5 \\ 2 \end{matrix}) = 44$ ,

$Π_{2} : r = (\begin{matrix} 0 \\ 0 \\ 3 \end{matrix}) + λ (\begin{matrix} 2 \\ 0 \\ - 3 \end{matrix}) + μ (\begin{matrix} 1 \\ 1 \\ 1 \end{matrix})$ .

Find the shortest distance between the two planes $Π_{1}$ and $Π_{2}$ .

3-11-4-ib-hl-aa-short-dist-two-planes-we-solution-2

DP IB Maths: AA HL

Revision Notes

3.11.4 Shortest Distances with Planes

Shortest Distance Between a Line and a Plane

How do I find the shortest distance between a given point on a line and a plane?

How do I find the shortest distance between a plane and a line parallel to the plane?

Exam Tip

Worked Example

Shortest Distance Between Two Planes

How do I find the shortest distance between two parallel planes?

How do I find the shortest distance from a given point on a plane to another plane?

Exam Tip

Worked Example

Author: Amber

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