How to compute the parametric equations of a line

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How to compute the parametric equations of a line passing through two points. How to compute the equations of a plane given its normal vector and a point on the plane. How to compute the equations of a plane passing through three points. How to compute vector and scalar projections. How to calculate the distance from a point to a plane and from a point to a line. How to identify the level and contour curves of a real-valued functions of two variables. How to compute limits of a vector-valued function of one variable. How to compute limits of a real-valued function of two variables or determine if such limit does not exists. How to determine if a real-valued function of two variables is continuous at a given point. How to compute the partial derivatives of a real-valued function. 2
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SUMMARY A vector is an arrow between two points in the cartesian plane or space. Given two points A = ( x 1 , y 1 , z 1 ) and B = ( x 2 , y 2 , z 2 ) then the vector between them is ~ AB = h x 2 - x 1 , y 2 - y 1 , z 2 - z 1 i . The norm (or magnitude) of a vector ~v = h a, b, c i is k ~v k = a 2 + b 2 + c 2 . Vectors can be added and scaled as follows: h a, b, c i + h a 0 , b 0 , c 0 i = h a + a 0 , b + b 0 , c + c 0 i , λ · h a, b, c i = h λa, λb, λc i . The dot product between two vectors ~v = h a, b, c i and ~w = h a 0 , b 0 , c 0 i is defined by ~v · ~w = aa 0 + bb 0 + cc 0 . It can be shown that if θ is the angle between the vectors ~v and ~w then ~v · ~w = k ~v kk ~w k cos θ. It then follows that ~v and ~w are orthogonal if and only if ~v · ~w = 0. There is another product of vectors whose output is a vector. This is called the cross product: ~v × ~w = det i j k a b c a 0 b 0 c 0 = ( bc 0 - b 0 c ) i - ( ac 0 - a 0 c ) j + ( ab 0 - a 0 b ) k , where i = h 1 , 0 , 0 i , j = h 0 , 1 , 0 i and k = h 0 , 0 , 1 i . From the properties of determinants it follows that ~v × ~w is orthogonal to both ~v and ~w , i.e., ~v · ( ~v × ~w ) = 0 and ~w · ( ~v × ~w ) = 0 . It is also easy to check that ~v × ~w = - ~w × ~v . 3
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The norm of the cross product is k ~v × ~w k = k ~v kk ~w k sin θ, where θ is the angle between ~v and ~w . Therefore ~v and ~w are parallel if and only if ~v × ~w = ~ 0. The area of the parallelogram generated by the vectors ~v and ~w is A = k ~v × ~w k .
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Christopher Reinemann
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