# Find the Plane Through (1,2,-3),(3,5,-3) Parallel to the Line Through (1,-1,1),(-2,-2,-2) (1,2,-3) , (3,5,-3) , (1,-1,1) , (-2,-2,-2) , , ,
Given points and , find a plane containing points and that is parallel to line .
First, calculate the direction vector of the line through points and . This can be done by taking the coordinate values of point and subtracting them from point .
Replace the , , and values and then simplify to get the direction vector for line .
Calculate the direction vector of a line through points and using the same method.
Replace the , , and values and then simplify to get the direction vector for line .
The solution plane will contain a line that contains points and and with the direction vector . For this plane to be parallel to the line , find the normal vector of the plane which is also orthogonal to the direction vector of the line . Calculate the normal vector by finding the cross product x by finding the determinant of the matrix .
Calculate the determinant of .
Set up the determinant by breaking it into smaller components.
The determinant of is .
The determinant of a matrix can be found using the formula .
Simplify the determinant.
Simplify each term.
Multiply by .
Multiply by .
Simplify the expression.
Move to the left of .
Since the matrix is multiplied by , the determinant is .
The determinant of is .
The determinant of a matrix can be found using the formula .
Simplify the determinant.
Move to the left of .
Simplify by multiplying through.
Apply the distributive property.
Multiply.
Multiply by .
Multiply by .
Solve the expression at point since it is on the plane. This is used to calculate the constant in the equation for the plane.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Simplify by adding and subtracting.