Given the line (x+2)/2=(y-7)/3 = (z+8)/-3, and the plane 2x+3y-4z-7=0
Determine the point of intersection, and the angle of intersection, if they exist
These multiple posts of similar topics are starting to look like a homework dump. Reiny has helped quite a few times; have you no ideas of your own to show?
You MUST know how to change the given line equation into its
parametric form.
Once you have done that, sub in the x, y, and z values into the plane equation.
You will have an equation in terms of the parameter, which lets you find the x, y, and z
of the intersection point.
let me know what you get
hint for the angle:
dot product of the normal and the direction vector of the line will give you what angle?
Make a sketch
When someone is in an advanced math class, but then says s/he doesn't know some pretty basic things, my first thought is that the person probably had trouble with the class before that, too.
My best suggestion is that you go to https://www.khanacademy.org/ and click on Courses at the top left. Choose from among the Get ready for courses and complete them before dealing with calculus. I think* you should start with Get ready for Algebra 1, then … Algebra 2, then … precalculus. Math is like climbing a ladder. If you miss steps, you'll fall.
*A math tutor may have different ideas for you.
To find the point of intersection, we'll solve the given line and plane equations simultaneously.
1. Start with the line equation:
(x+2)/2 = (y-7)/3 = (z+8)/-3
2. Let's take the first two ratios and equate them:
(x+2)/2 = (y-7)/3
3. Cross-multiplying, we get:
3(x+2) = 2(y-7)
3x + 6 = 2y - 14
3x - 2y = -20 (Equation 1)
4. Now let's take the second and third ratios and equate them:
(y-7)/3 = (z+8)/-3
5. Cross-multiplying, we get:
-3(y-7) = -3(z+8)
-3y + 21 = -3z - 24
3y + 3z = 45 (Equation 2)
6. We have a system of two equations (Equation 1 and Equation 2):
3x - 2y = -20
3y + 3z = 45
7. To eliminate y, we'll multiply Equation 1 by 3 and Equation 2 by 2:
9x - 6y = -60
6y + 6z = 90
8. Adding the equations together, we have:
9x + 6z = 30 (Equation 3)
9. Now, let's solve Equations 3 and Equation 2 simultaneously:
9x + 6z = 30
3y + 3z = 45
10. Rearranging Equation 2, we get:
z = 15 - y
11. Substitute the value of z in Equation 3:
9x + 6(15 - y) = 30
9x + 90 - 6y = 30
9x - 6y = -60
12. This equation is the same as Equation 1. It means the line and plane are coincident (overlap) since they have the same equation.
13. So, the point of intersection is infinite, and the angle of intersection cannot be determined.
In conclusion, the line and plane coincides, therefore there is an infinite number of points of intersection, and the angle of intersection cannot be determined.