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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector PlanesShortest Distances with Planes

Shortest Distances with Planes (DP IB Analysis & Approaches (AA)): Revision Note

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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 

Examiner Tips and Tricks

This skill is not explicitly stated in the syllabus guide. However, I have seen this come up in Paper 2 in the November 2022 exams. It was worth 5 marks!

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 Tips and Tricks

  • 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

Examiner Tips and Tricks

This skill is not explicitly stated in the syllabus guide. However, I have seen this come up in Paper 1 in the May 2024 exams. It was worth 7 marks, but there was a hint in the question to help you.

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 Tips and Tricks

  • 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

    Author: Amber

    Expertise: Maths

    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.