MATH 230 Midterm: Math 230 Exam1-Solutions
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|u v| = 0 (iv) (iv) (g) (e) (f) (i) Solution: the vectors u and v are perpendicular if the angle between them is equal to /2, point in the same direction when = 0, and point in the opposite direction when = . Now use the formulas u v = |u| |v| cos and |u v| = |u| |v| sin which in our case take the form u v = cos and. |u v| = sin since |u| = |v| = 1. Find an equation of the sphere passing through the points a = (0, 2, 1) and. B = (4, 0, 3) with its center at the midpoint of the line segment ab. Solution: the center c of the sphere has the coordinates. C = (0 + 4)/2, (2 + 4)/2, (1 + 3)/2 = (2, 3, 2). The radius r of the sphere is equal to. 2 and the equation of the sphere is given by.
