Angles Between Lines & Planes (DP IB Maths: AA HL)

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Angle Between Line & Plane

What is meant by the angle between a line and a plane?

  • When you find the angle between a line and a plane you will be finding the angle between the line itself and the line on the plane that creates the smallest angle with it
    • This means the line on the plane directly under the line as it joins the plane
  • It is easiest to think of these two lines making a right-triangle with the normal vector to the plane
    • The line joining the plane will be the hypotenuse
    • The line on the plane will be adjacent to the angle
    • The normal will the opposite the angle

How do I find the angle between a line and a plane?

  • You need to know:
    • A direction vector for the line (b)
      • This can easily be identified if the equation of the line is in the form bold italic r equals bold italic a plus lambda bold italic b
    • A normal vector to the plane (n)
      • This can easily be identified if the equation of the plane is in the form bold italic r times bold italic n equals bold italic a times bold italic n
  • Find the acute angle between the direction of the line and the normal to the plane
    • Use the formula cos alpha equals fraction numerator vertical line bold italic b times bold italic n vertical line over denominator vertical line bold italic b vertical line vertical line bold italic n vertical line end fraction
      • The absolute value of the scalar product ensures that the angle is acute
  • Subtract this angle from 90° to find the acute angle between the line and the plane
    • Subtract the angle from straight pi over 2 if working in radians

3-11-3-angle-between-a-line-and-a-plane-diagram-1

Examiner Tip

  • Remember that if the scalar product is negative your answer will result in an obtuse angle
    • Taking the absolute value of the scalar product will ensure that you get the acute angle as your answer

Worked example

Find the angle in radians between the line L with vector equation bold italic r equals open parentheses 2 minus lambda close parentheses bold italic i plus open parentheses lambda plus 1 close parentheses bold italic j plus open parentheses 1 minus 2 lambda close parentheses bold italic k and the plane capital pi with Cartesian equation x minus 3 y plus 2 z equals 5.

3-11-3-ib-hl-aa-angle-line-and-plane-we-solution-1

Angle Between Two Planes

How do I find the angle between two planes?

  • The angle between two planes is equal to the angle between their normal vectors
    • It can be found using the scalar product of their normal vectors
    • cos theta equals fraction numerator bold italic n subscript bold 1 times bold italic n subscript bold 2 over denominator open vertical bar bold italic n subscript bold 1 close vertical bar open vertical bar bold italic n subscript bold 2 close vertical bar end fraction
  • If two planes Π1 and Π2 with normal vectors n1 and n2 meet at an angle then the two planes and the two normal vectors will form a quadrilateral
    • The angles between the planes and the normal will both be 90°
    • The angle between the two planes and the angle opposite it (between the two normal vectors) will add up to 180°

3-11-3-ib-hl-aa-angle-between-two-planes-diagram-2

Examiner 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

Find the acute angle between the two planes which can be defined by equations begin mathsize 16px style capital pi subscript 1 colon blank 2 x minus y plus 3 z equals blank 7 end style and begin mathsize 16px style capital pi subscript 2 colon blank x plus 2 y minus z equals 20 end style.

3-11-3-ib-hl-aa-angle-between-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.