Find the equation of the plane containing the point and parallel to the plane
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Ask by Dunn Chavez. in the United States

Jan 13,2025

Question

Find the equation of the plane containing the point $$ (-1,-4,-8) $$ and parallel to the plane $$ 7 x-6 y-7 z=0 $$.

UpStudy AI · Accepted Answer

To find the equation of the plane that contains the point $$(-1, -4, -8)$$ and is parallel to the plane $$7x - 6y - 7z = 0$$, follow these steps:

1. **Identify the Normal Vector:**
   - The given plane $$7x - 6y - 7z = 0$$ has a normal vector $$\mathbf{n} = (7, -6, -7)$$.
   - Since the desired plane is parallel to this one, it shares the same normal vector.

2. **Use the Point-Normal Form:**
   - The general equation of a plane with normal vector $$(A, B, C)$$ that passes through a point $$(x_0, y_0, z_0)$$ is:
     $$
     A(x - x_0) + B(y - y_0) + C(z - z_0) = 0
     $$
   - Plugging in the normal vector $$(7, -6, -7)$$ and the point $$(-1, -4, -8)$$:
     $$
     7(x + 1) - 6(y + 4) - 7(z + 8) = 0
     $$
   
3. **Simplify the Equation:**
   - Expanding the equation:
     $$
     7x + 7 - 6y - 24 - 7z - 56 = 0
     $$
   - Combine like terms:
     $$
     7x - 6y - 7z - 73 = 0
     $$
   - Rearranging gives:
     $$
     7x - 6y - 7z = 73
     $$

**Final Equation of the Plane:**
$$
7x - 6y - 7z = 73
$$

Answer: After simplifying, an equation for the plane is 7 x – 6 y – 7 z = 73
The equation of the plane is $$7x - 6y - 7z = 73$$.

Find the equation of the plane containing the point and parallel to the plane
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Find the equation of the plane containing the point and parallel to the plane .