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Engineering Calculus Notes 45

Engineering Calculus Notes 45 - determine a line the slope...

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1.3. LINES IN SPACE 33 1.3 Lines in Space Parametrization of Lines A line in the plane is the locus of a “linear” equation in the rectangular coordinates x and y Ax + By = C where A , B and C are real constants with at least one of A and B nonzero. A geometrically informative version of this is the slope-intercept formula for a non-vertical line y = mx + b (1.14) where the slope m is the tangent of the angle the line makes with the horizontal and the y -intercept b is the ordinate (signed height) of its intersection with the y -axis. Unfortunately, neither of these schemes extends verbatim to a three-dimensional context. In particular, the locus of a “linear” equation in the three rectangular coordinates x , y and z Ax + By + Cz = D is a plane , not a line. Fortunately, though, we can use vectors to implement the geometric thinking behind the point-slope formula ( 1.14 ). This formula separates two pieces of geometric data which together
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Unformatted text preview: determine a line: the slope re±ects the direction (or tilt) of the line, and then the y-intercept distinguishes between the various parallel lines with a given slope by specifying a point which must lie on the line. A direction in 3-space cannot be determined by a single number, but it is naturally speci²ed by a nonzero vector, so the three-dimensional analogue of the slope of a line is a direction vector −→ v = a −→ ı + b −→ + c −→ k to which it is parallel. 6 Then, to pick out one among all the lines parallel to −→ v , we specify a basepoint P ( x ,y ,z ) through which the line is required to pass. 6 In general, we do not need to restrict ourselves to unit vectors; any nonzero vector will do....
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