8.1 Vector and Parametric Equations of a Line in R2

Example2expressingtheequationsoflinesusingvectors

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Unformatted text preview: line. x b) A line passes through the points 1 1 3 A( , ‐3) and B( , ). 4 2 2 Determine a direction vector for this line, and write it using integer components. Example 2 Expressing the Equations of Lines Using Vectors Find the parametric and vector equations of a line through A(xo, yo) with direction vector m = (a, b) in R2. OP = OPo + PoP l y OP = (xo, yo) + tm r = ro+ tm (x, y) = (xo, yo) + t(a, b) x Parametric Equations of Line l: x = x o + at y = yo + bt Vector Equation of Line l: (x, y) = (xo, yo) + t(a, b) r = ro + tm Note: (xo, yo) is any point on the line m = (a, b) is not unique, ∵any parallel vector will also work ∴we use tm, with t as the parameter, t εR 2 8.1 Vector and Parametric Equations of a Line in R2.notebook March 08, 2010 Example 3 Reasoning about the vector and parametric equations of a line Find the vector and parametric equations of a line through: a) A(‐1, ‐5) and having direction vector m = (3, 2). b) Sketch the line. c) Find 2 other points on the line. y x Example 4 Connecting vector and parametric equations with two points on a line Find vector and parametric equations of the line through A(3, ‐1) and Q(‐1, 2). 3 8.1 Vector and Parametric Equations of a Line in R2.notebook March 08, 2010 Example 4 Find the vector equation of a perpendicular line. Consider the line r = (‐4, 3) + t(2, ‐1), t εR a) Find the direction vector and sketch the line. b) Find the intercepts. c) Which point corresponds to the parameter value t = 1? d) Determine a vector equation for a line that is perpendicular to the given line, and passes through the point P(‐5, 3). y x Homework page 433 ‐ 434 #1‐6, 8‐12, 14 4...
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