# Algebra Examples

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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 adding numbers.

Add and 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 .

Add and to get .

Multiply by to get .

Simplify each term.

Multiply by to get .

Rewrite as .

Subtract from to get .

Multiply by to get .

Add and to get .

Divide each term by and simplify.

Divide each term in by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by to get .

Solve the equation for .

Simplify each term.

Cancel the common factor of .

Rewrite.

Write as a fraction with denominator .

Factor out the greatest common factor .

Cancel the common factor.

Rewrite the expression.

Simplify.

Multiply by to get .

Multiply and to get .

Multiply by to get .

Move the negative in front of the fraction.

Subtract from to get .

Solve the equation for .

Simplify each term.

Multiply by to get .

Simplify .

Multiply by to get .

Multiply by to get .

Add and to get .

Solve the equation for .

Simplify each term.

Multiply by to get .

Simplify .

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 .