Find the vector eqn of the plane through the points (2,1,-1) and (-1,3,4) and perpendicular to the plane x-2y+4z=10
To find the vector equation of a plane passing through two given points and perpendicular to a given plane, we will follow these steps:
Step 1: Find the normal vector of the given plane.
Step 2: Find the direction vector of the plane passing through the given points.
Step 3: Use the normal vector and direction vector to write the vector equation of the desired plane.
Let's perform these steps in order:
Step 1: Find the normal vector of the plane x - 2y + 4z = 10.
A plane equation can be written in the form ax + by + cz = d, where (a, b, c) is the normal vector to the plane.
Comparing the given plane equation x - 2y + 4z = 10 with the standard form, we have:
a = 1, b = -2, c = 4
Therefore, the normal vector to the given plane is n = (1, -2, 4).
Step 2: Find the direction vector of the plane passing through the points (2, 1, -1) and (-1, 3, 4).
We can obtain the direction vector by subtracting the coordinates of one point from the other point.
Let's subtract the coordinates of the first point from the coordinates of the second point:
Direction vector = (-1, 3, 4) - (2, 1, -1)
= (-1 - 2, 3 - 1, 4 - (-1))
= (-3, 2, 5)
So, the direction vector of the plane passing through the given points is v = (-3, 2, 5).
Step 3: Write the vector equation of the desired plane using the normal vector (n) and the direction vector (v).
The vector equation of a plane passing through a point (x_0, y_0, z_0) and having a normal vector (a, b, c) is:
r = (x_0, y_0, z_0) + t(a, b, c) + s(u, v, w)
In this case, we can take one of the given points as the "point" and the direction vector as the "a, b, c" in the equation.
Let's select the first point (2, 1, -1) as the "point":
r = (2, 1, -1) + t(1, -2, 4) + s(-3, 2, 5)
Therefore, the vector equation of the plane passing through the points (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x - 2y + 4z = 10 is:
r = (2, 1, -1) + t(1, -2, 4) + s(-3, 2, 5)
Note: The variables t and s are parameters that can take any real values to generate different points on the plane.