Introduction to Finding the Equation of a Plane
In three-dimensional space, finding the equation of a plane that passes through a point and contains a given line is a fundamental concept in vector and linear algebra. This article explains the process step-by-step, providing a detailed guide that aligns with Google's search engine optimization (SEO) standards.
Understanding the Problem
We are given a point (P_0 (8, 0, -2)) and a line defined by the parametric equations:
(x 5 - 3t) (y 1 4t) (z 7 4t)The goal is to find the equation of the plane that passes through (P_0) and contains the entire line.
Step-by-Step Solution
Step 1: Identify Key Points and Vectors
The given line can be visualized as passing through a point on the line, say (P_1 (5, 1, 7)), with a direction vector (vec{d} (-3, 4, 4)).
Step 2: Determine the Second Direction Vector
The direction vector from (P_0 (8, 0, -2)) to (P_1 (5, 1, 7)) is:
(vec{v} (5 - 8, 1 - 0, 7 - (-2)) (-3, 1, 9))
Step 3: Calculate the Normal Vector to the Plane
The normal vector (vec{n}) to the plane can be found by taking the cross product of (vec{v}) and (vec{d}):
(vec{n} vec{v} times vec{d} begin{vmatrix} mathbf{i} mathbf{j} mathbf{k} -3 1 9 -3 4 4 end{vmatrix} (-3 cdot 4 - 4 cdot 9)mathbf{i} - (-3 cdot 4 - (-3) cdot 9)mathbf{j} (-3 cdot 4 - (-3) cdot 1)mathbf{k})
(vec{n} (-12 - 36, 12 27, -12 3) (-48, 39, -9))
Step 4: Form the Plane Equation
The general form of the plane equation is (ax by cz d). To find (a, b, c, d), we use the point (P_0 (8, 0, -2)) and the normal vector (vec{n} (-48, 39, -9)):
( -48(8) 39(0) - 9(-2) -238 )
Thus, the equation of the plane is:
(-48x 39y - 9z -238)
Verification
For (P_0 (8, 0, -2)): (-48(8) 39(0) - 9(-2) -384 18 -238) For a point on the line (t 1) (( (2, 5, 11) )): (-48(2) 39(5) - 9(11) -96 195 - 99 -238) For another point on the line (t -1) (( (8, -3, 3) )): This point is the same as (P_0) and similarly verifies the equation.Conclusion
The equation of the plane that passes through the point (P_0 (8, 0, -2)) and contains the line defined by (x 5 - 3t), (y 1 4t), and (z 7 4t) is:
(-48x 39y - 9z -238)