Linear Algebra Examples

Find the Intersection of the Line Perpendicular to Plane 1 Through the Origin and Plane 2 y=3x+2 , x-4y=9
,
Step 1
Subtract from both sides of the equation.
Step 2
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 3
Find the normal vectors for each plane and determine if they are perpendicular by calculating the dot product.
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Step 3.1
is . Find the normal vector from the plane equation of the form .
Step 3.2
is . Find the normal vector from the plane equation of the form .
Step 3.3
Calculate the dot product of and by summing the products of the corresponding , , and values in the normal vectors.
Step 3.4
Simplify the dot product.
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Step 3.4.1
Remove parentheses.
Step 3.4.2
Simplify each term.
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Step 3.4.2.1
Multiply by .
Step 3.4.2.2
Multiply by .
Step 3.4.2.3
Multiply by .
Step 3.4.3
Simplify by adding and subtracting.
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Step 3.4.3.1
Subtract from .
Step 3.4.3.2
Add and .
Step 4
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 5
Substitute the expression for , , and into the equation for .
Step 6
Solve the equation for .
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Step 6.1
Simplify .
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Step 6.1.1
Combine the opposite terms in .
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Step 6.1.1.1
Subtract from .
Step 6.1.1.2
Add and .
Step 6.1.2
Multiply by .
Step 6.1.3
Subtract from .
Step 6.2
Divide each term in by and simplify.
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Step 6.2.1
Divide each term in by .
Step 6.2.2
Simplify the left side.
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Step 6.2.2.1
Cancel the common factor of .
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Step 6.2.2.1.1
Cancel the common factor.
Step 6.2.2.1.2
Divide by .
Step 6.2.3
Simplify the right side.
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Step 6.2.3.1
Move the negative in front of the fraction.
Step 7
Solve the parametric equations for , , and using the value of .
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Step 7.1
Solve the equation for .
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Step 7.1.1
Remove parentheses.
Step 7.1.2
Remove parentheses.
Step 7.1.3
Simplify .
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Step 7.1.3.1
Multiply .
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Step 7.1.3.1.1
Multiply by .
Step 7.1.3.1.2
Combine and .
Step 7.1.3.1.3
Multiply by .
Step 7.1.3.2
Add and .
Step 7.2
Solve the equation for .
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Step 7.2.1
Remove parentheses.
Step 7.2.2
Remove parentheses.
Step 7.2.3
Simplify .
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Step 7.2.3.1
Multiply by .
Step 7.2.3.2
Subtract from .
Step 7.3
Solve the equation for .
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Step 7.3.1
Remove parentheses.
Step 7.3.2
Remove parentheses.
Step 7.3.3
Simplify .
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Step 7.3.3.1
Multiply .
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Step 7.3.3.1.1
Multiply by .
Step 7.3.3.1.2
Multiply by .
Step 7.3.3.2
Add and .
Step 7.4
The solved parametric equations for , , and .
Step 8
Using the values calculated for , , and , the intersection point is found to be .