Let L be the line passing through the point P=(−3, −1, −5) with direction vector →d=[4, 0, 4]T, and let T be the plane defined by 4x−y−3z = 16. Find the point Q where L and T intersect.
please help..
Sarah, I did not check back on your post til now, sorry
so sub back in:
4x - y - 3z = 16
4(-3 + 4t) - (-1) - 3(-5 + 4t) = 16
-12 + 16t + 1 + 15 - 12t = 16
4t = 12
t = 3
x = -3+12=9 , y = -1, z =-5+12 = 7
so the point of intersection is (9,-1,7)
I have no idea how you got your answer
parametric equation of your line:
x = -3 + 4t
y = -1
z = -5 + 4t
sub those into the equation of the plane and solve for t
then get x, y, and z
or am I doing something wrong?
To find the point where the line L intersects with the plane T, we need to determine the coordinates of the point Q.
1. Find the Cartesian equation of the line L:
- Given the point P(-3, -1, -5) on the line L.
- The direction vector of the line L is →d = [4, 0, 4]ᵀ.
- The Cartesian equation of the line L can be expressed as:
x = -3 + 4t,
y = -1 + 0t,
z = -5 + 4t,
where t is a parameter.
2. Substitute the equations of x, y, and z into the equation of the plane T:
4x - y - 3z = 16
Substituting x, y, and z from the line L equation:
4(-3 + 4t) - (-1) - 3(-5 + 4t) = 16.
3. Solve the equation obtained in step 2 for the parameter t:
4(-3 + 4t) + 1 - 3(-5 + 4t) = 16
-12 + 16t + 1 + 15 - 12t = 16
4t + 4 = 16
4t = 16 - 4
4t = 12
t = 3.
4. Substitute the value of t into the line L equation to find the coordinates of point Q:
x = -3 + 4t = -3 + 4(3) = -3 + 12 = 9
y = -1 + 0t = -1 + 0(3) = -1
z = -5 + 4t = -5 + 4(3) = -5 + 12 = 7.
Therefore, the point Q where the line L intersects the plane T is Q(9, -1, 7).
x=7/4
y=15
z=1/12
is this correct?