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8-6: Vectors and Parametric Equations
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Objectives Write vector and parametric equations of lines.
Graph parametric equations.
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Applications For objects moving in a straight line, there is a more useful way of writing equations for the line describing the object's path using vectors.
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Vector Equations
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Vector Equation of a Line
A line through P1(x1,y1) parallel to vector is defined by the set of points P1(x1,y1) and P2(x2,y2) such that for some real number t. Therefore,
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Example Write a vector equation describing a line passing through P1(-9,3) and parallel to An equation like above can be used to describe the coordinates for a point on the line for any value of t (often time).
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Parametric Equations of a Line
A line through P1(x1,y1) that is parallel to vector has the following parametric equations where t is any real number.
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Example Find the parametric equations for a line parallel to and passing through the point at (4,3). Then make a table of values and graph. Evaluating the parametric equations for a value of t gives the coordinates of the position of the object after t units of time. t x y -1 1 8 4 3 7 -2
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Example Write the parametric equations of y=1/2 x + 7.
(parametric t: independent and x&y: dependent) (in given equation: x is independent and y is dependent) t=x y=1/2 t + 7 If you make a table of values and graph these parametric equations, you'll see that it's the same graph as y = 1/2 x + 7. parametric equations
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Example Write an equation in slope-intercept form of the line whose parametric equations are x= -2 + t and y=4-3t. (Solve both equations for t and set them equal to each other.)
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Homework 8-6: p. 524 # mult of 3, 37, 11
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