the plane is -7x-5y+3z=23 and the parametric equation is <-3,-7,-8>t+ <-8,2,3> - find the intersection equation

To find the intersection of the plane \( -7x - 5y + 3z = 23 \) with the line given by the parametric equations, we need to substitute the parametric equations of the line into the plane equation. 

### Step 1: Write the parametric equations of the line.
The line is given in parametric form as:
\[
x = -3t - 8
\]
\[
y = -7t + 2
\]
\[
z = -8t + 3
\]

### Step 2: Substitute the parametric equations into the plane equation.
Substitute \( x = -3t - 8 \), \( y = -7t + 2 \), and \( z = -8t + 3 \) into the plane equation \( -7x - 5y + 3z = 23 \):

\[
-7(-3t - 8) - 5(-7t + 2) + 3(-8t + 3) = 23
\]

### Step 3: Simplify the equation.
First, distribute the constants:

\[
21t + 56 + 35t - 10 - 24t + 9 = 23
\]

Now, combine like terms:

\[
(21t + 35t - 24t) + (56 - 10 + 9) = 23
\]

\[
32t + 55 = 23
\]

### Step 4: Solve for \( t \).
Isolate \( t \):

\[
32t = 23 - 55
\]

\[
32t = -32
\]

\[
t = -1
\]

### Step 5: Find the point of intersection.
Substitute \( t = -1 \) back into the parametric equations of the line to find the point of intersection:

\[
x = -3(-1) - 8 = 3 - 8 = -5
\]

\[
y = -7(-1) + 2 = 7 + 2 = 9
\]

\[
z = -8(-1) + 3 = 8 + 3 = 11
\]

### Final Answer: 
The point of intersection is \((-5, 9, 11)\).

Would you like further details on this solution? Here are some related questions:

1. How do you find the equation of a plane given three points?
2. What is the geometric interpretation of the dot product in relation to the angle between two vectors?
3. How do you convert a plane equation from standard form to normal form?
4. What is the significance of the parametric equations of a line in 3D space?
5. How can you determine if two lines are skew, parallel, or intersecting in 3D space?

**Tip:** When solving for the intersection of a line and a plane, ensure that the line isn't parallel to the plane, which would imply no intersection (unless the line lies within the plane).

Learn how to find the intersection of the plane -7x-5y+3z=23 with the parametric equation <-3,-7,-8>t+ <-8,2,3>. Follow a detailed step-by-step solution involving substitution of parametric equations into the plane equation and solving for t. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation