# Algebra Examples

, , ,

Step 1

Given points and , find a plane containing points and that is parallel to line .

Step 2

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 .

Step 3

Replace the , , and values and then simplify to get the direction vector for line .

Step 4

Calculate the direction vector of a line through points and using the same method.

Step 5

Replace the , , and values and then simplify to get the direction vector for line .

Step 6

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 .

Step 7

Step 7.1

Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.

Step 7.1.1

Consider the corresponding sign chart.

Step 7.1.2

The cofactor is the minor with the sign changed if the indices match a position on the sign chart.

Step 7.1.3

The minor for is the determinant with row and column deleted.

Step 7.1.4

Multiply element by its cofactor.

Step 7.1.5

The minor for is the determinant with row and column deleted.

Step 7.1.6

Multiply element by its cofactor.

Step 7.1.7

The minor for is the determinant with row and column deleted.

Step 7.1.8

Multiply element by its cofactor.

Step 7.1.9

Add the terms together.

Step 7.2

Evaluate .

Step 7.2.1

The determinant of a matrix can be found using the formula .

Step 7.2.2

Simplify the determinant.

Step 7.2.2.1

Simplify each term.

Step 7.2.2.1.1

Multiply by .

Step 7.2.2.1.2

Multiply .

Step 7.2.2.1.2.1

Multiply by .

Step 7.2.2.1.2.2

Multiply by .

Step 7.2.2.2

Add and .

Step 7.3

Evaluate .

Step 7.3.1

The determinant of a matrix can be found using the formula .

Step 7.3.2

Simplify the determinant.

Step 7.3.2.1

Simplify each term.

Step 7.3.2.1.1

Multiply by .

Step 7.3.2.1.2

Multiply by .

Step 7.3.2.2

Subtract from .

Step 7.4

Evaluate .

Step 7.4.1

The determinant of a matrix can be found using the formula .

Step 7.4.2

Simplify the determinant.

Step 7.4.2.1

Simplify each term.

Step 7.4.2.1.1

Multiply by .

Step 7.4.2.1.2

Multiply by .

Step 7.4.2.2

Add and .

Step 7.5

Simplify each term.

Step 7.5.1

Move to the left of .

Step 7.5.2

Multiply by .

Step 8

Step 8.1

Simplify each term.

Step 8.1.1

Multiply by .

Step 8.1.2

Multiply by .

Step 8.1.3

Multiply by .

Step 8.2

Simplify by subtracting numbers.

Step 8.2.1

Subtract from .

Step 8.2.2

Subtract from .

Step 9

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

Step 10

Multiply by .