# 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.
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 .
Multiply by .
Multiply by .
Simplify by adding numbers.
Add and .
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
Solve the equation for .
Simplify .
Combine the opposite terms in .
Add and .
Subtract from .
Simplify each term.
Multiply by .
Rewrite as .
Multiply by .
Add and .
Divide each term in by and simplify.
Divide each term in by .
Simplify the left side.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify the right side.
Move the negative in front of the fraction.
Step 6
Solve the parametric equations for , , and using the value of .
Solve the equation for .
Remove parentheses.
Remove parentheses.
Simplify .
Simplify each term.
Cancel the common factor of .
Move the leading negative in into the numerator.
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Multiply by .
Move the negative in front of the fraction.
Subtract from .
Solve the equation for .
Remove parentheses.
Remove parentheses.
Simplify .
Multiply .
Multiply by .
Multiply by .
Add and .
Solve the equation for .
Remove parentheses.
Remove parentheses.
Simplify .
Multiply .
Multiply by .
Multiply by .
Add and .
The solved parametric equations for , , and .
Step 7
Using the values calculated for , , and , the intersection point is found to be .
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