shortest-distances—planes

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Shortest distance between a point & 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 $bold r equals bold a plus lambda bold b$ 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, on $size 16px l$
  - 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 $capital pi$ 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

Worked Example

The plane $capital pi$ has equation <img alt=”bold r times open parentheses table row 2 row cell negative 1 end cell row 1 end table close parentheses equals 6″ data-mathml='<math ><semantics><mstyle mathsize=”16px”><mi mathvariant=”bold”>r</mi><mo>⋅</mo><mfenced separators=”|”><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mn>6</mn></mstyle><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″}</annotation></semantics></math>’ height=”69″ role=”math” 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Further Maths: Core Pure -Edexcel-A Level

shortest-distances—planes

Shortest distance between a point & a plane

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

Worked Example

Responses