Algebra Examples

Find the Intersection of the Line Perpendicular to Plane 1 Through the Origin and Plane 2
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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 .
Find the normal vectors for each plane and determine if they are perpendicular by calculating the dot product.
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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.
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Remove parentheses.
Simplify each term.
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Multiply by to get .
Multiply by to get .
Multiply by to get .
Simplify by adding numbers.
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Add and to get .
Add and to get .
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 .
Substitute the expression for , , and into the equation for .
Solve the equation for .
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Simplify the left side.
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Simplify each term.
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Multiply by to get .
Add and to get .
Multiply by to get .
Simplify each term.
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Multiply by to get .
Rewrite as .
Subtract from to get .
Multiply by to get .
Add and to get .
Divide each term by and simplify.
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Divide each term in by .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by to get .
Solve the parametric equations for , , and using the value of .
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Solve the equation for .
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Simplify each term.
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Cancel the common factor of .
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Rewrite.
Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Simplify.
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Multiply by to get .
Multiply and to get .
Multiply by to get .
Move the negative in front of the fraction.
Subtract from to get .
Solve the equation for .
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Simplify each term.
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Multiply by to get .
Simplify .
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Multiply by to get .
Multiply by to get .
Add and to get .
Solve the equation for .
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Simplify each term.
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Multiply by to get .
Simplify .
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Multiply by to get .
Multiply by to get .
Add and to get .
The solved parametric equations for , , and .
Using the values calculated for , , and , the intersection point is found to be .
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