# Algebra Examples

Find the Intersection of the Line Perpendicular to Plane 1 Through the Origin and Plane 2
,
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
Find the normal vectors for each plane and determine if they are perpendicular by calculating the dot product.
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
Step 2.4.3.1
Step 2.4.3.2
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
Solve the equation for .
Step 5.1
Simplify .
Step 5.1.1
Combine the opposite terms in .
Step 5.1.1.1
Step 5.1.1.2
Subtract from .
Step 5.1.2
Simplify each term.
Step 5.1.2.1
Multiply by .
Step 5.1.2.2
Rewrite as .
Step 5.1.2.3
Multiply .
Step 5.1.2.3.1
Multiply by .
Step 5.1.2.3.2
Multiply by .
Step 5.1.3
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
Divide by .
Step 6
Solve the parametric equations for , , and using the value of .
Step 6.1
Solve the equation for .
Step 6.1.1
Remove parentheses.
Step 6.1.2
Simplify .
Step 6.1.2.1
Multiply by .
Step 6.1.2.2
Subtract from .
Step 6.2
Solve the equation for .
Step 6.2.1
Remove parentheses.
Step 6.2.2
Simplify .
Step 6.2.2.1
Multiply by .
Step 6.2.2.2
Step 6.3
Solve the equation for .
Step 6.3.1
Remove parentheses.
Step 6.3.2
Simplify .
Step 6.3.2.1
Multiply by .
Step 6.3.2.2