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.
Tap for more steps...
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.
Tap for more steps...
Step 2.4.1
Remove parentheses.
Step 2.4.2
Simplify each term.
Tap for more steps...
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.
Tap for more steps...
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
Solve the equation for .
Tap for more steps...
Step 5.1
Simplify .
Tap for more steps...
Step 5.1.1
Combine the opposite terms in .
Tap for more steps...
Step 5.1.1.1
Add and .
Step 5.1.1.2
Subtract from .
Step 5.1.2
Simplify each term.
Tap for more steps...
Step 5.1.2.1
Multiply by .
Step 5.1.2.2
Rewrite as .
Step 5.1.2.3
Multiply .
Tap for more steps...
Step 5.1.2.3.1
Multiply by .
Step 5.1.2.3.2
Multiply by .
Step 5.1.3
Add and .
Step 5.2
Divide each term in by and simplify.
Tap for more steps...
Step 5.2.1
Divide each term in by .
Step 5.2.2
Simplify the left side.
Tap for more steps...
Step 5.2.2.1
Cancel the common factor of .
Tap for more steps...
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.
Tap for more steps...
Step 5.2.3.1
Divide by .
Step 6
Solve the parametric equations for , , and using the value of .
Tap for more steps...
Step 6.1
Solve the equation for .
Tap for more steps...
Step 6.1.1
Remove parentheses.
Step 6.1.2
Simplify .
Tap for more steps...
Step 6.1.2.1
Multiply by .
Step 6.1.2.2
Add and .
Step 6.2
Solve the equation for .
Tap for more steps...
Step 6.2.1
Remove parentheses.
Step 6.2.2
Subtract from .
Step 6.3
Solve the equation for .
Tap for more steps...
Step 6.3.1
Remove parentheses.
Step 6.3.2
Simplify .
Tap for more steps...
Step 6.3.2.1
Multiply by .
Step 6.3.2.2
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 .
Enter YOUR Problem
Mathway requires javascript and a modern browser.