, , ,

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 .

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.

Add and .

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 .

Add and .

Simplify each term.

Multiply by .

Multiply by .

Multiply by .

Simplify by adding and subtracting.

Add and .

Subtract from .

Add the constant to find the equation of the plane to be .

Multiply by .

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)