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Practice Problems Test 1 Part 1 Name: Date:

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Solve the following problems.

(1) Let A and B be two points in R2 or R3. Show that C is in the segment from A to B if there are numbers x and y such that x~ A + y~ B = ~ C and x + y = 1.

(2) Let A, B, and C be three points in R2 or R3. Show that C can be written as xA + y~ B = C with x + y = 1 if C is in the segment from A to B.

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(3) Let ~ A and ~ B be two vectors in the xy-plane making angles α and β with the x-axis. Show the following (a) ~ A = cosα~ i + sinα~ j ; ~ B = cosβ~ i + sinβ~ j; (b) cos(α−β) = cosαcosβ + sinαsinβ. (4) Let ~ A, ~ B, ~ C, and ~ D be four vectors. If ~ B − ~ A = 2(~ B − ~ C) show that the intersection point of the two line segments joining A and D and B and C trisect these segments.

(5) Prove in four dierent ways the points (1,2,−3),(3,0,−5),(−4,7,2) are collinear.

(6) Prove in two dierent ways that the angle inscribed in a semicircle is a right angle using vectors and do the proof without using vectors.

(7) Let y1 = mx + 5 and y2 = nx − 3 be two lines. (a) Use the dot product of vectors to prove that if y1 is perpendicular to y2, then mn = −1. (b) Use vectors to show that if y1 is parallel to y2, then m = n.

(8) Assume ~ A = ~ A1 + ~ A2 and ~ A1 · ~ B = ~ 0, and also that ~ A2 is parallel to ~ B. Show that ~ A2 =~ A·~ B ||~ B||2~ B (9) Given A = (2,−1,1) and B = (1,−3,−5), nd a point C such that A, B, C are the vertices of a right triangle. Give two solutions.

(10) Use vectors to prove that the median to the base of an isosceles triangle is perpendicular to the base. Hint: Use problems 1 and 2.

(11) The points A = (3,5,−2), B = (1,0,4), and C = (3,5,−2) are the vertices of a triangle. Determine the following: (a) The length of the sides of the triangle 4ABC. (b) The midpoint of the sides of 4ABC. (c) The lengths of the three medians of 4ABC. (12) (a) Use the dot product to nd a vector that is perpendicular to (2,1,−2) and to (−1,0,1). (b) Solve the same problem using the cross product. Are the two vectors found parallel?


(13) Prove that the points (3,4,1), (−3,2,−6), and (5,−3,7) are the vertices of an isosceles triangle and nd the angles of it. Calculate the area of this triangle in three dierent ways.

(14) (a) Prove that the points (−1,4,3), (−3,10,7), and (0,1,8) are the vertices of a right triangle. (b) Determine the length of the median to the hypotenuse of this right triangle and compare it with the length of the hypotenuse. Can you generalize this result?

(15) Use vectors to prove Pythagoras Theorem.

(16) (a) Use the dot product to decide whether the triangle with vertices (1,−3,−2), (2,0,−4), and (6,−2,−5) is right-angled. (b) Solve the problem without using the dot product.

(17) For any two vectors ~ u and ~ v prove that |~ u · ~ v| ≤ ||~ u||||~ v||. This inequal-ity is called Cauchy-Schwartz Inequality.

(18) For any two vectors ~ u and ~ v prove, using the Cauchy-Schwartz Inequality, that ||~ u + ~ v||≤||~ u||+||~ v||. This is called the triangle inequality. (19) Use vectors to prove that the distance D from a point P = (x0,y0) to the line ax + by + c = 0 is D = |ax0 + by0 + c| √a2 + b2 . (20) Show that if ~ A+ ~ B and ~ A−~ B are orthogonal vectors, then the vectors ~ A and ~ B must have the same norm.

(21) If θ is the angle between the vectors ~ u and ~ v, show that

proj~ u~ v·proj~ v~ u = (~ u·~ v)cos2 θ. (22) Given A = (2,−2,1) and B = (1,15,1), nd the number z such that the point C = (5,1,z) together with the points A and B are the vertices of a right triangle. There are two dierent answers.

(23) (a) Use the distance formula to prove that the midpoint of the line segment from P1 = (x1,y1,z1) to P2 = (x2,y2,z2) is x1 + x2 2 , y1 + y2 2 , z1 + z2 2 . (b) Usevectors to prove (a).

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