# Algebra Examples

,

Step 1

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 .

Step 2

Step 2.1

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

Step 2.2

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

Step 2.3

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

Step 2.4

Simplify the dot product.

Step 2.4.1

Remove parentheses.

Step 2.4.2

Simplify each term.

Step 2.4.2.1

Multiply by .

Step 2.4.2.2

Multiply by .

Step 2.4.2.3

Multiply by .

Step 2.4.3

Simplify by adding numbers.

Step 2.4.3.1

Add and .

Step 2.4.3.2

Add and .

Step 3

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 .

Step 4

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

Step 5

Step 5.1

Simplify .

Step 5.1.1

Combine the opposite terms in .

Step 5.1.1.1

Add and .

Step 5.1.1.2

Subtract from .

Step 5.1.2

Simplify each term.

Step 5.1.2.1

Rewrite as .

Step 5.1.2.2

Multiply by .

Step 5.1.3

Add and .

Step 5.2

Divide each term in by and simplify.

Step 5.2.1

Divide each term in by .

Step 5.2.2

Simplify the left side.

Step 5.2.2.1

Cancel the common factor of .

Step 5.2.2.1.1

Cancel the common factor.

Step 5.2.2.1.2

Divide by .

Step 5.2.3

Simplify the right side.

Step 5.2.3.1

Move the negative in front of the fraction.

Step 6

Step 6.1

Solve the equation for .

Step 6.1.1

Remove parentheses.

Step 6.1.2

Remove parentheses.

Step 6.1.3

Simplify .

Step 6.1.3.1

Simplify each term.

Step 6.1.3.1.1

Multiply .

Step 6.1.3.1.1.1

Multiply by .

Step 6.1.3.1.1.2

Combine and .

Step 6.1.3.1.1.3

Multiply by .

Step 6.1.3.1.2

Move the negative in front of the fraction.

Step 6.1.3.2

Subtract from .

Step 6.2

Solve the equation for .

Step 6.2.1

Remove parentheses.

Step 6.2.2

Remove parentheses.

Step 6.2.3

Simplify .

Step 6.2.3.1

Multiply .

Step 6.2.3.1.1

Multiply by .

Step 6.2.3.1.2

Multiply by .

Step 6.2.3.2

Add and .

Step 6.3

Solve the equation for .

Step 6.3.1

Remove parentheses.

Step 6.3.2

Remove parentheses.

Step 6.3.3

Simplify .

Step 6.3.3.1

Multiply .

Step 6.3.3.1.1

Multiply by .

Step 6.3.3.1.2

Multiply by .

Step 6.3.3.2

Add and .

Step 6.4

The solved parametric equations for , , and .

Step 7

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