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Calculus Homework Assignment 4 Class: Student Number: Name: 1. a. Prove that the midpoint of the line segment from P1 (x1 , y1 , z1 ) to P2 (x2 , y2 , z2 ) is ( x1 + x2 y1 + y2 z1 + z2 , , 2 2 2 ) 3. a. Draw the vectors a = ⟨3, 2⟩, b = ⟨2, −1⟩, and c = ⟨7, 1⟩. b. Show, by means of a sketch, that there are scalars s and t such that c = sa + tb. c. Find the exact values of s and t. [§13.2 #39(a),(b),(d)] b. Find the lengths of the medians of the triangle with vertices A(1, 2, 3), B(−2, 0, 5), and C(4, 1, 5). [§13.1 #19] 2. Find a unit vector that has the same direction as the given vector. a. ⟨−4, 2, 4⟩ b. 8i − j + 4k [§13.2 #22, 23] 4. a. Show that the vector ortha b = b−proja b is orthogonal to a. (It is called an orthogonal projection of b.) b. Under what circumstances is compa b =compb a, and what circumstances is proja b =projb a? [§13.3 #41, 44] (Over Please) Calculus Homework Assignment 4 5. Use a scalar projection to show that the distance from a point P1 (x1 , y1 ) to the line ax + by + c = 0 is |ax1 + by1 + c| √ a2 + b2 7. a. Prove that (a − b) × (a + b) = 2(a × b). a·c b·c b. Prove that (a × b) · (c × d) = a · d b · d [§13.4 #45, 48] Use this formula to find the distance from the point (−2, 3) to the line 3x − 4y + 5 = 0. [§13.3 #49] 6. Let P be a point not on the line L that 8. Suppose a ̸= 0. passes trough the points Q and R. Show that a. If a · b = a · c, does it follow that b = c? the distance d from the point P to the line L is b. If a × b = a × c, does it follow that b = c? c. If a · b = a · c and a × b = a × c, does it follow |a × b| d= that b = c? [§13.4 #49] |a| −−→ −−→ where a = QR and b = QP . Use the formula to find the distance from the point P (1, 1, 1) to the line through Q(0, 6, 8) and R(−1, 4, 7). [§13.4 #43]