# Algebra Examples

,

Move to the left side of the equation because it contains a variable.

Move to the left side of the equation because it contains a variable.

Move to the right side of the equation because it does not contain a variable.

To find the intersection of the line through a point perpendicular to plane and plane :

1. Find the normal vectors of plane and plane where the normal vectors are and . Check to see if the dot product is 0.

2. Create a set of parametric equations such that , , and .

3. Substitute these equations into the equation for plane such that and solve for .

4. Using the value of , solve the parametric equations , , and for to find the intersection .

is . Find the normal vector from the plane equation of the form .

is . Find the normal vector from the plane equation of the form .

Calculate the dot product of and by summing the products of the corresponding , , and values in the normal vectors.

Simplify the dot product.

Remove parentheses.

Simplify each term.

Multiply by to get .

Multiply by to get .

Multiply by to get .

Simplify by subtracting numbers.

Subtract from to get .

Add and to get .

Next, build a set of parametric equations ,, and using the origin for the point and the values from the normal vector for the values of , , and . This set of parametric equations represents the line through the origin that is perpendicular to .

Substitute the expression for , , and into the equation for .

Simplify the left side.

Simplify each term.

Multiply by to get .

Rewrite as .

Multiply by to get .

Simplify by adding terms.

Subtract from to get .

Subtract from to get .

Divide each term by and simplify.

Divide each term in by .

Simplify the left side of the equation by cancelling the common factors.

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Move the negative one from the denominator of .

Simplify the expression.

Multiply by to get .

Rewrite as .

Divide by to get .

Solve the equation for .

Multiply by to get .

Subtract from to get .

Solve the equation for .

Multiply by to get .

Add and to get .

Solve the equation for .

Simplify each term.

Multiply by to get .

Multiply by to get .

Add and to get .

The solved parametric equations for , , and .

Using the values calculated for , , and , the intersection point is found to be .