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Please advise on a suitable method. Thanks in advance!

Question: The planes m and n have equations 3x+y-2z=10 & x-2y+2z=5 respectively.

The line L has equation r=4i+2j+k+(lamda)(i+j+2k).

i) Show that L is parallel to m.

ii) Calculate the acute angle between the planes m and n.

iii) A point P lies on the line L. The perpendicular distance of P from the plane n is equal to 2. Find the position vectors of the two possible positions of P.

Memo:

i) - answer not specified explicitly on given mark scheme -

ii) 74.5 degrees

iii) 7i+5j+7k from lamda=3 ; and 3i+j-k from lamda=-1

[Cambridge International Advanced Level, Mathematics, Paper 3 Pure Mathematics 3, 9709/31, October/November 2018, Question 10]

Manon Blackbeak
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  • And ? What do you think ? –  Jun 17 '21 at 09:03
  • @YvesDaoust At this point, I can comfortably get to a plane equation, but I'm having some trouble on how to get the normal vector from a plane equation (so, working backwards then). – Manon Blackbeak Jun 17 '21 at 10:08
  • The normal is "written" in the equation ! –  Jun 17 '21 at 10:11
  • @YvesDaoust Direction vector of line L = i+j+2k. I'm guessing that the relationship between direction vector L and normal vector m (which I'm having trouble finding) is probably going to come down to scalar product of both not = 0 ? Again, I'm having trouble with finding the normal vectors for both m and n, and therefor don't know what to substitute into cos(theta)=scalar product of two vectors/product of moduli of vectors. – Manon Blackbeak Jun 17 '21 at 10:14
  • Ahh, so I get it! Can't believe I missed that. – Manon Blackbeak Jun 17 '21 at 10:17

1 Answers1

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Hint:

  • You know the equations of planes $$m: 3x+y-2z=10 \quad \text{and} \quad n: x-2y+2z=5.$$ So, can you find the normal vectors for the planes $m$ and $n$ respectively?
  • You know that the line has equation, can you find the director vector of line $\ell$?
  • What is the relation between vector director of line $\ell$ and normal vector for the plane $m$?
  • What is the relation between normal vector for the plane $m$ with normal vector for the plane $n$? Do you know as calculate the angle between vectors?