Likewise a line l in three dimensional space is

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Unformatted text preview: ine and the direction of the line (its slope or angle of inclination) are given. The equation of the line can then be written using the point-slope form. Likewise, a line L in three-dimensional space is determined when we know a point P0 x 0 , y0 , z0 on L and the direction of L. In three dimensions the direction of a line is conveniently described by a vector, so we let v be a vector parallel to L. Let P x, y, z be an arbitrary point on L and let r0 and r be the position vectors of P0 and P (that is, they have representations OP0 and OP ). If a is the vector with representation P0 P , as in Figure 1, A A A then the Triangle Law for vector addition gives r r0 a. But, since a and v are parallel vectors, there is a scalar t such that a t v. Thus r 1 z t=0 t>0 L t<0 r¸ x FIGURE 2 y r0 tv which is a vector equation of L. Each value of the parameter t gives the position vector r of a point on L. In other words, as t varies, the line is traced out by the tip of the vector r. As Figure 2 indicates, positive values of t correspond to points on L that lie on one side of P0 , whereas negative values of t correspond to points that lie on the other side of P0 . If the vector v that gives the direction of the line L is written in component form as v a, b, c , then we have t v ta, tb, tc . We can also write r x, y, z and r0 x 0 , y0 , z0 , so the vector equation (1) becomes x, y, z x0 ta, y0 tb, z0 tc 5E-13(pp 858-867) 1/18/06 11:27 AM Page 859 SECTION 13.5 EQUATIONS OF LINES AND PLANES ❙❙❙❙ 859 Two vectors are equal if and only if corresponding components are equal. Therefore, we have the three scalar equations: 2 x x0 at y y0 z bt z0 ct where t . These equations are called parametric equations of the line L through the point P0 x 0 , y0 , z0 and parallel to the vector v a, b, c . Each value of the parameter t gives a point x, y, z on L. |||| Figure 3 shows the line L in Example 1 and its relation to the given point and to the vector that gives its direction. z (a) Here r0 5, 1, 3 tion (1) becomes r¸ v=i+4 j-2 k x (...
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