Question

The plane p has equation 3x+2y+4z=13. A second plane q is perpendicular to p and has equation ax+y+z=4, where a is a constant.
(i) Find the value of a.
(ii) The line with equation r=j−k+ λ (i+2j+2k)meets the plane p at the point A and the plane q at the point B. Find the length of AB.

Answer 1

if 2 planes are perpendicular, their normals are perpendicular
the normal of plane 1 is <3,2,4>
the normal of plane 2 is <a,1,1>

Answer 2

and ?

Answer 3

do you know how to test if two vectors are perpendicular ? does "dot product" ring any bells?

Answer 4

yeah

Answer 5

can you find a?

Answer 6

a=2

Answer 7

huh ?

Answer 8

are you saying <3,2,4> dot <2,1,1> = 0
all the numbers are positive.: 6+2+4 does not look promising

Answer 9

-2

Answer 10

the dot product is 0 if the two vectors are perpendicular.
with a=2, the dot product is not 0

Answer 11

I would do
<3,2,4> dot <a,1,1>= 0
3a + 2+ 4 =0
3a+6= 0
3a= -6
a= -2

Answer 12

yeah ive got it

Answer 13

to find the intersection of a line with a plane, start with
P=j−k+ a (i+2j+2k)
and
<3,2,4> dot P = 13

I would write the equation of the line as
P= <0,1,-1> + a<1,2,2>
and use that in the equation for the plane
<3,2,4> dot ( <0,1,-1> + a<1,2,2> ) = 13
solve for a (i.e. lambda, but easier to type)
and then find P using the line equation