IBMaths: AA HLDPRevision Notes3. Geometry & Trigonometry3.11 Vector Planes3.11.4 Shortest Distances with Planes

Shortest Distances with Planes (DP IB Maths: AA HL)

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Shortest Distance Between a Line and a Plane

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

  • The shortest distance from any point to a plane will always be the perpendicular distance from the point to the plane
  • Given a point, P with position vector p and a plane capital pi with equation begin mathsize 16px style bold r times bold n equals d end style
    • STEP 1: Find the vector equation of the line perpendicular to the plane that goes through the point, P
      • This will have the position vector of the point, P, and the direction vector n
      • bold r equals bold p plus lambda bold n
    • STEP 2: Find the value of lambda at the point of intersection of this line with capital pi by substituting the equation of the line into the equation of the plane
    • STEP 3: Find the distance between the point and the point of intersection
      • Substitute lambda into the equation of the line to find the coordinates of the point on the plane closest to point P
      • Find the distance between this point and point P
      • As a shortcut, this distance will be equal to vertical line lambda bold n vertical line 

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
  • You can follow the same steps above
  • 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 begin mathsize 16px style l subscript 1 end style with equation bold r equals bold a plus lambda bold b and a plane capital pi parallel to l subscript 1 with equation begin mathsize 16px style bold r times bold n equals d end style
    • Where n is the normal vector to the plane
    • STEP 1: Find the equation of the line begin mathsize 16px style l subscript 2 end style perpendicular to l subscript 1 and capital pi going through the point a in the form bold r equals bold a plus mu bold n
    • STEP 2: Find the point of intersection of the line l subscript 2 and capital pi
    • 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

Examiner 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 capital pi has equation bold r times open parentheses table row 2 row cell negative 1 end cell row 1 end table close parentheses equals 6.

The line begin mathsize 16px style L end style has equation bold r equals open parentheses table row 2 row 3 row 1 end table close parentheses plus s open parentheses table row 1 row cell negative blank 2 end cell row 4 end table close parentheses.

The point P space left parenthesis negative 2 comma space 11 comma space minus 15 right parenthesis lies on the line begin mathsize 16px style L end style.

Find the shortest distance between the point P and the plane capital pi.

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 capital pi subscript 1 with equation bold r times bold n equals d and a plane capital pi subscript 2 with equation bold r equals bold a plus lambda bold b plus blank mu bold 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 bold r times bold n equals d to find the coordinates of the point where the line meets capital pi subscript 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, capital pi subscript 1, to another plane, capital pi subscript 2 will be the perpendicular distance from the point to capital pi subscript 2
    • STEP 1: Use the given coordinates of the point P on capital pi subscript 1 and the normal to the plane capital pi subscript 2 to find the vector equation of the line through P that is perpendicular to capital pi subscript 1
    • STEP 2: Find the point of intersection of this line with the plane capital pi subscript 2
    • STEP 3: Find the distance between the two points of intersection

Examiner 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:

capital pi subscript 1 blank colon space space bold r times open parentheses table row 3 row cell negative 5 end cell row 2 end table close parentheses space equals space 44,

capital pi subscript 2 blank colon space space bold r bold space equals space open parentheses table row 0 row 0 row 3 end table close parentheses space plus space lambda open parentheses table row 2 row 0 row cell negative 3 end cell end table close parentheses space plus space mu open parentheses table row 1 row 1 row 1 end table close parentheses.

Find the shortest distance between the two planes begin mathsize 16px style capital pi subscript 1 end style and capital pi subscript 2.

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

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    Amber

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    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.