# Thread: Points, angles and lines of intersection.

1. ## Points, angles and lines of intersection.

a) Determine where the line L : x = (-1, 8, 9)+t(3,-9,-4) and the plane
E : -2x1 + 6x2 + 3x3 + 19 = 0 intersect and calculate the angle between
them.
b) Given the two planes
E : . = 2.x + 3x2 + 7x3 + 2 = 0, E’ : x=(3,0,0)+ s(-2,0,0)+t(3,0,1)
find the line of intersection and the angle between them.

2. What have you tried?

Originally Posted by mamt6
a) Determine where the line L : x = (-1, 8, 9)+t(3,-9,-4) and the plane
E : -2x1 + 6x2 + 3x3 + 19 = 0 intersect and calculate the angle between
them.
Note that we can write the line as, $L = \left<-1 + 3t, 8 - 9t, 9 - 4t \right>$.

That is, $x_1 = -1 + 3t,~~x_2 = 8-9t,~~x_3 = 9 - 4t$

You can use that to find the point of intersection. As for the angle of inetersection, use the formula

$\bold{a} \cdot \bold{b} = \| \bold a\|\|\bold b\| \cos \theta$

Here $\bold a$ and $\bold b$ are vectors and $\theta$ is the angle between them. (Hint: one of your vectors should be lying in the plane )

b
) Given the two planes
E : . = 2.x + 3x2 + 7x3 + 2 = 0, E’ : x=(3,0,0)+ s(-2,0,0)+t(3,0,1)
find the line of intersection and the angle between them.
The normal vector for $E'$ is given by $\left< -2, 0, 0 \right> \times \left< 3, 0 , 1 \right>$.

The angle between the planes is the same as the angle between their normal vectors. Use the formula I gave you just above.

As for the line of intersection, we can find it if we find a point on the line and a vector in the direction of the line. The latter can be found by taking the cross product of the two normal vectors (Why?). For the former, set $x_3 = 0$ in both planes. This will tell you on what lines the planes intersect with the xy-plane. Then just find the intersection point of those two lines (remember, the z-coordinate of those points is 0), that will give you a point on the line of intersection.

Now see if you can get anywhere with those hints

3. i understand how to do a) but im still not sure about b) im pretty sure im meant to convert the parametric equation (E') into a normal equation in the form of E but im not sure how i got about doing it.