Algebra Examples

Find the Plane Through (1,2,-3),(3,5,-3) Parallel to the Line Through (1,-1,1),(-2,-2,-2)
, , ,
Given points and , find a plane containing points and that is parallel to line .
First, calculate the direction vector of the line through points and . This can be done by taking the coordinate values of point and subtracting them from point .
Replace the , , and values to get and then simplify to get the direction vector for line .
Calculate the direction vector of a line through points and using the same method.
Replace the , , and values to get and then simplify to get the direction vector for line .
The solution plane will contain a line that contains points and and with the direction vector . For this plane to be parallel to the line , find the normal vector of the plane which is also orthogonal to the direction vector of the line . Calculate the normal vector by finding the cross product x by finding the determinant of the matrix .
Calculate the determinant of .
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Set up the determinant by breaking it into smaller components.
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Multiply by to get .
Multiply by to get .
Simplify the expression.
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Add and to get .
Move to the left of the expression .
Multiply by to get .
Since the matrix is multiplied by , the determinant is .
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Multiply by to get .
Multiply by to get .
Move to the left of the expression .
Multiply by to get .
Simplify by multiplying through.
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Apply the distributive property.
Multiply.
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Multiply by to get .
Multiply by to get .
Add and to get .
Solve the expression at point since it is on the plane. This is used to calculate the constant in the equation for the plane.
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Simplify each term.
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Multiply by to get .
Multiply by to get .
Multiply by to get .
Simplify by adding and subtracting.
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Add and to get .
Subtract from to get .
Add the constant to find the equation of the plane to be .
Simplify each term.
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Multiply by to get .
Multiply by to get .
Multiply by to get .
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