Question

Suppose that the line l is represented by r(t) = <11 +2t, 23 +6t, 31 +8t> and the plane P is represented by 3x +4y +5z = 35.

1. Find the intersection of the line l and the plane P.Write your answer as a point (a,b,c) where a,b,c are numbers.

2. Find the cosine of the angle Theta between the line l and the normal vector of the plane P.

Answer 1

to find the intersection:

3x +4y +5z = 35

now sub the vector coordinates in terms of t into x,y,z

3(11 + 2t) +4(23 + 6t) + 5(31 + 8t) = 35

now solve for t.

once you get a value for t, sub it back into

r(t) = <11 +2t, 23 +6t, 31 +8t>

to find the point in (a, b, c) form

Answer 2

for part b:

The normal line vector is <3, 4, 5>

now use the formula

a . b = ||a||||b||cosine(theta)

where <a> is <3,4,5>

and <b> is the vector at the point where it intersects the plane at time t

Answer 3

b is basically the answer from part 1?