{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Likewise a line l in three dimensional space is

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 (...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online