ࡱ > ! '` bjbj$$ 4* F F p 8 4 , $ \ A C C C C C C $ x h R g g | X A A V @ P pa8@ V Y , 0 e x 2 ( 2 2 8 g g 6 X Section 8.1 Vector and Parametric Equations of a Line in R To find the vector and/or parametric equations of a line in R, we must be given either two distinct points or one point and a direction vector*. A direction vector is defined to be a nonzero vector EMBED Equation.3 parallel (i.e., collinear) to the given line. The x and y components of the direction vector EMBED Equation.3 are called the direction numbers (in other words, the direction numbers of EMBED Equation.3 are a and b). Example 1 from page 427 A line passing through EMBED Equation.3 has EMBED Equation.3 as its direction vector. Sketch this line. A line passes through the points EMBED Equation.3 and EMBED Equation.3 . Determine a direction vector for this line, and write it using integer components. If we know a point on a line, as well as a direction vector for the line, then we can state the vector equation of the line by using its position vector and its direction vector. The position vector is the vector that extends from the origin to the known point. The direction vector is the vector that extends from one known point on the line to another known point on the line (just like we found in part b. of Example 1 above) Vector Equation of the Line: EMBED Equation.3 , EMBED Equation.3 Parametric Equations of the Line: EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 (copy diagram from page 428) Example 2 from page 429 Determine the vector and parametric equations of a line passing through point EMBED Equation.3 with direction vector EMBED Equation.3 . Sketch the line, and determine the coordinates of four points on the line. Is either point EMBED Equation.3 or point EMBED Equation.3 on this line? Example 3 from page 430 Determine vector and parametric equations for the line containing the points EMBED Equation.3 and EMBED Equation.3 . What are the coordinates of the point where this line crosses the x-axis? Can the equation EMBED Equation.3 , EMBED Equation.3 , also represent the line containing the points E and F? Example 4 from page 431 Determine a vector equation for the line that is perpendicular to EMBED Equation.3 , EMBED Equation.3 , and passes through point EMBED Equation.3 * this is analogous to what you learned in Grade 9, that to find the equation of a line you need two points or one point and the slope of the line. L R T X Y } ~ 6 7 J K ξίξΓuf jP h7t CJ UVaJ h7t h7t 5>* h7t 5>* j h?