Practice Problems For Final Math 5B Chapter 1: Vectors, Matrices, and Applications Summary: Practice Problems For Final Math 5B Chapter 1: Vectors, Matrices, and Applications Practice Problem: Write an equation for the plane that contains the points (2, 0, -3), (-4, -5, 2), and (0, 3, -4) in the form ax + by + cz = d. Solution: Let v = (-4, -5, 2) - (2, 0, -3) = (-6, -5, 5) and w = (0, 3, -4) - (2, 0, -3) = (-2, 3, -1). Then v × w = i(5 - 15) - j(6 + 10) + k(-18 - 10) = (-10, -16, -28). We can choose n to be any vector in the same direction as v × w so let n = (5, 8, 14). Then the plane has the form 5x + 8y + 14z = d. Substituting the point (2, 0, -3) for (x, y, z) and solving for d gives d = 10 + 0 + (-42) = -32. So the plane has the equation 5x + 8y + 14z = -32 . Practice Problem: Find a parametric form for the line passing through the point (1, 2) in the direction (3, 4), which we will call c1(t). Set c1(t) equal to (x, y) and eliminate t to get the line into y = mx + b form. Now find a different parametrization c2(t) of the same line such that c2(0) = (-2, -2) and c2(2) = (-5, -6). Solution: c1(t) = (1, 2) + t(3, 4) = (1 + 3t, 2 + 4t). Setting (x, y) = (1 + 3t, 2 + 4t) yields x = 1 + 3t and y = 2 + 4t. Solving the former equation for t yields t = (x - 1)/3. Substituting this into the second equation then gives us y = 4 3 Collections: Mathematics