B find two other points on the line solution l 513 r

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Unformatted text preview: a) Find a vector equation and parametric equations for the line that passes through the point 5, 1, 3 and is parallel to the vector i 4 j 2 k. (b) Find two other points on the line. SOLUTION L (5, 1, 3) EXAMPLE 1 5i j 3 k and v r 5i j r 5 ti y or 3k i ti 1 4j 2 k, so the vector equa- 4j 2k 4t j 3 2t k 4t z 3 Parametric equations are FIGURE 3 x 5 t y 1 2t (b) Choosing the parameter value t 1 gives x 6, y 5, and z point on the line. Similarly, t 1 gives the point 4, 3, 5 . 1, so 6, 5, 1 is a The vector equation and parametric equations of a line are not unique. If we change the point or the parameter or choose a different parallel vector, then the equations change. For instance, if, instead of 5, 1, 3 , we choose the point 6, 5, 1 in Example 1, then the parametric equations of the line become x 6 t y 5 z 4t 1 2t Or, if we stay with the point 5, 1, 3 but choose the parallel vector 2 i arrive at the equations x 5 2t y 1 z 8t 3 8j 4 k, we 4t In general, if a vector v a, b, c is used to describe the direction of a line L, then the numbers a, b, and c are called direction numbers of L. Since any vector parallel to v could also be used, we see that any three numbers proportional to a, b, and c could also be used as a set of direction numbers for L. Another way of describing a line L is to eliminate the parameter t from Equations 2. If none of a, b, or c is 0, we can solve each of these equations for t, equate the results, and obtain 3 x x0 a y y0 b z z0 c 5E-13(pp 858-867) ❙❙❙❙ 860 1/18/06 11:27 AM Page 860 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE These equations are called symmetric equations of L . Notice that the numbers a, b, and c that appear in the denominators of Equations 3 are direction numbers of L, that is, components of a vector parallel to L. If one of a, b, or c is 0, we can still eliminate t. For instance, if a 0, we could write the equations of L as x y x0 b This means that L lies in the vertical plane x |||| Figure 4 shows the line L in Example 2 and the point P where it intersects the xy-plane. z 1 B x z0 c x 0. EXAMPLE 2 (a) Find parametric equations and symmetric equati...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas.

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