# Algebra Examples

Find the Plane Through (1,2,-3),(3,5,-3) Parallel to the Line Through (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 to get 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 to get 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 to get .
Multiply by to get .
Simplify the expression.
Move to the left of the expression .
Multiply by to get .
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.
Simplify each term.
Multiply by to get .
Multiply by to get .
Move to the left of the expression .
Multiply by to get .
Simplify by multiplying through.
Apply the distributive property.
Multiply.
Multiply by to get .
Multiply by to get .
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 to get .
Multiply by to get .
Multiply by to get .
Subtract from to get .
Add the constant to find the equation of the plane to be .
Simplify each term.
Multiply by to get .
Multiply by to get .
Multiply by to get .

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