Formula for Slope of 3-D LineDate: 03/12/97 at 10:33:13 From: Cory and Greg Subject: Slope of three-dimensional line Thank you for answering our question about finding the length of a line in three dimensions. Now we would like to know the formula to find the slope of a three-dimensional line. We searched through textbooks and tried to adapt the formula, but with no success. Date: 03/12/97 at 12:37:40 From: Doctor Rob Subject: Re: Slope of three-dimensional line There is no direct analogue of the idea of slope in two dimensions. The subject you are discussing is analytic geometry of three dimensions. The following facts should help a little. A linear equation in x, y, and z, such as ax + by + cz = d, is the equation of a plane, not a line. Such equations can be put into one standard form by dividing by sqrt(a^2+b^2+c^2). The resulting coefficients of x, y, and z have the property that the sum of their squares is 1. Another standard form is gotten by dividing by d, and writing the equation as x/(d/a) + y/(d/b) + z/(d/c) = 1. From this form you can read off the intercepts with the x-, y-, and z-axes: (d/a, 0, 0) is the x-intercept, (0, d/b, 0) is the y-intercept, and (0, 0, d/c) is the z-intercept (provided all of a, b, c, and d are nonzero). One way of regarding the "slope" of a plane is to write down a unit vector which is perpendicular to it, called the normal vector. It is given by (a*I + b*J + c*K)/sqrt(a^2+b^2+c^2), where I, J, and K are the unit vectors in the x, y, and z directions. The coefficients of I, J, and K in this expression are called the direction cosines of the vector, because they are the cosines of the angles between the vector and the x-, y-, and z-axes, respectively. A line is specified as the intersection of two nonparallel planes. This means you need two linear equations in x, y, and z to determine a line. There are several standard forms for the equations of a line, but a commonly used one is x - x0 y - y0 z - z0 ------ = ------ = ------ a b c Here (x0, y0, z0) is a point on the line, and the numbers a, b, and c determine the direction along the line: the vector a*I = b*J + c*K is parallel to the line. (Note: This form only works when the line is not parallel to any of the xy-, xz-, or yz-planes, i.e., when neither a, b, or c is zero). In some sense, the direction cosines are the closest analogue to the slope. In two dimensions, they are just the cosine of the inclination, which is the angle with the x-axis, and the cosine of its complement, which is the sine of the inclination. The slope is the ratio of these two, the tangent of the inclination. There is no exact analogue because there is no "ratio" of three direction cosines, or of any three numbers. If you need further explanation, write to us again. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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