Vector Equations Trig 6.12 Obj: Find the vector equation parallel to a given vector and through a given point

White Board Review If vector AB has displacement (2, 3) and A = (1, -1), what is B? If vector CD has displacement (2, 3) and D = (-2, 5), what is C? If P = (0, 3) and Q = (1, -2), find the displacement of vector PQ. Find the displacement of vector QP. Let O = (0, 0), A = (1, 3), B = (3, 2), C = (5, 1) Determine which of the following vectors are equivalent: OA, OB, OC, AB, AC, BC

Vector Equation of a Line A line through P1 parallel to the vector a is defined by the set of points P2 such that vector P1P2 = ta. Standard form: X = (x, y) + t(a1, a2) Example: Write a vector equation describing a line passing through (1, 4) and parallel to the vector a = (-1, 2). X = (1, 4) + t(-1, 2)

Point Testers Consider the line with vector equation X = (2, 5) + t(-4, 2). Test whether the point (5, 7) is on the line. 2 + -4t = 5 t = -3/4 5 + 2t = 7 t = 1 Since the times are not equal, the point is not on the line. Test whether the point (4, 4) is on the line.

Application The variable t used in vector equations is not a physical coordinate, it is a parameter, usually representing time, that lets you give the x and y coordinates in relation to time. Two test cars start racing along straight lines in a flat desert. Suppose Car 1 starts at (0, 0) and travels with constant velocity (25, 40). Car 2 starts at the same time at (10, 25) and travels with velocity vector (30, 45). Do the paths intersect? Do the cars crash?

Solution First, see if the paths intersect. Set the positions equal to each other and solve for time. (0, 0) + t(25, 40) = (10, 25) + s(30, 45) 0 + 25t = 10 + 30s 0 + 40t = 25 + 45s -30s + 25t = 10 -45s + 40t = 25 Solve for s and t. Is there a solution? Are the times equal?

Continued. . . Find the intersection point for the two cars’ paths. Plug in the solutions s = 3 and t = 4. (0, 0) + 4(25, 40) = (100, 160) (10, 25) + 3(30, 45) = (100, 160) The two cars intersect at the point (100, 160).

Practice Write a vector equation of the line that passes through the point (-4, -11) and is parallel to the vector (-3, 8). Consider the line X = (1, 0) + t(1, 2). Find a Cartesian equation of the line. Consider the line X = (1, 2) + t(3, 4). Test whether the points (4, 6), (-5, -5) and (13, 18) are on the line. Find a vector equation through the points (-2, 3) and (3, 1).

Assignment 6.12 page 515 9, 13 – 16, 18