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Straight Lines
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The variable t behaves as a master controller for the variables x, y, and z, so the above system is a set of parametric equations for a straight line in Cartesian xyz space To completely define a straight, infinitely long line this way, we must let t vary throughout the entire set of real numbers, including t = 0 to fill the hole at the point (x0,y0,z0)
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An example Let s find the symmetric-form equation for the line L shown in Fig 18-2 As indicated in the drawing, L passes through the point
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P = ( 5, 4,3) and is parallel to the vector m = 3i + 5j 2k The direction numbers of L are the coefficients of the vector m, so we have a=3 b=5 c = 2
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L m = 3i + 5j 2k Each axis division equals 1 unit
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Line L and vector m are parallel
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P= ( 5, 4, 3)
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Figure 18-2
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What are the symmetric and parametric equations for line L
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Lines and Curves in Three-Space
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We are given a point P on the line L with the coordinates x0 = 5 y0 = 4 z0 = 3 The general symmetric-form equation for a line in Cartesian xyz space is (x x0)/a = (y y0)/b = (z z0)/c When we plug in the known values, we get the three-part equation [x ( 5)]/3 = [y ( 4)]/5 = (z 3)/( 2) which simplifies to (x + 5)/3 = (y + 4)/5 = (z 3)/( 2)
Another example Let s find a set of parametric equations for the line L shown in Fig 18-2 In this case, our work is easy We can take the values of x0, y0, z0, a, b, and c that we already know, and plug them into the generalized set of parametric equations
x = x0 + at y = y0 + bt z = z0 + ct The results are x = 5 + 3t y = 4 + 5t z = 3 2t
Are you confused
For any particular line in Cartesian xyz space, there are infinitely many valid ordered triples that can represent the direction numbers If a line has the direction numbers (2,3,4), then we can multiply all three entries by a real number other than 0 or 1, and we ll get another valid ordered triple of direction numbers For example, all of the following ordered triples represent the same line orientation as (2,3,4): (4,6,8) ( 2, 3, 4) (20,30,40) ( 20, 30, 40) (2p,3p,4p ) ( 2p, 3p, 4p )
Straight Lines
That s interesting, you say, but which direction numbers are the best In theory, it doesn t matter; any of the above ordered triples is as good as any other Nevertheless, from an esthetic point of view, it s a good idea to reduce an ordered triple of direction numbers so that the only common divisor is 1, and so that there is at most one negative element According to that standard, (2,3,4) are the preferred direction numbers
Here s a challenge!
Consider the following three-way equation that represents a straight line in Cartesian xyz space: 3x 6 = 4y 12 = 6z 24 Find a point on the line Determine the preferred direction numbers Based on that information, write down the direction vector as a sum of multiples of i, j, and k
Solution
Before we think about the direction numbers or any specific point on the line, let s try to get the equation into the standard symmetric form We can multiply the left-hand part of the equation by 4/4, the middle part by 3/3, and the right-hand part by 2/2 That gives us 4(3x 6)/4 = 3(4y 12)/3 = 2(6z 24)/2 Multiplying out the numerators, we get (12x 24)/4 = (12y 36)/3 = (12z 48)/2 We can factor out 12 from each of the numerators to obtain 12(x 2)/4 = 12(y 3)/3 = 12(z 4)/2 Dividing the entire equation through by 12 gives us the standard symmetric form (x 2)/4 = (y 3)/3 = (z 4)/2 We remember that the generalized symmetric equation for a straight line in Cartesian xyz space is (x x0)/a = (y y0)/b = (z z0)/c where (x0,y0,z0) are the coordinates of a specific point on the line, and a, b, and c are the direction numbers Comparing the symmetric-form equation we derived with the generalized form, we can see that x0 = 2 y0 = 3 z0 = 4