Plane equations

Question

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]

Answer 1

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?