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# **Vector Operations and Applications** **Problem 1:** Find the resultant of each pair of vectors using the triangle or parallelogram rule. Round the resultant to the nearest tenth of a centimeter. Determine its direction relative to the horizontal using a ruler and protractor. * **a:** a = (2,...

# **Vector Operations and Applications** **Problem 1:** Find the resultant of each pair of vectors using the triangle or parallelogram rule. Round the resultant to the nearest tenth of a centimeter. Determine its direction relative to the horizontal using a ruler and protractor. * **a:** a = (2, 4, -3), b = (-5, -7, 1), c = (8, 5, -9) * **Equation 1:** 2a + 5b - 3c = 12 * **Equation 2:** b - 6a + 2c = 13 **Problem 2:** Find the coordinate representation and length of AB, given the initial and terminal points of AB in each case. * **a:** A(2, 1), B(-1, 7) * **b:** A(1, -3), B(-5, 1) **Problem 3:** A soccer player kicks a ball at 30 m/s at an angle of 25° with the horizontal. The player runs at 4 m/s. What is the resultant velocity (magnitude and direction) of the ball? **Problem 4:** Hot air balloons. Twelve hot air balloons were launched. After several minutes, the coordinates of the first and second balloons are as follows: * Balloon 1: (20, 25, 30) * Balloon 2: (-29, 15, 10) **Questions**: * Find the distance between the two balloons. * If the third balloon is located at the midpoint between the first and second balloons, what are its coordinates? **Problem 5:** Find the unit vector in the direction of u in each case. * **a:** u = (6, -3) * **b:** u = (-1, 4) **Problem 6:** Find the dot product of vectors u and v in each case. Determine if they are orthogonal (perpendicular). * **a.** u = (-2, 4, 6), v = (3, 7, 12) * **b.** u = 9i + 5j + 11k, v = -5i - 7j - 6k * **c.** u = (2, 5), v = (-3, 2) * **d.** u = (4, 3), v = (6, 8) * **e.** u = 10i - 3j, v = i + 8j * **f.** u = (1, 3), v = (-4, 2) **Problem 7:** Find the cross product of vectors u and v in each case. Show that u × v is orthogonal to both u and v. * **a.** u = (1, 7, 3), v = (9, 4, 11) * **b.** u = 6i + 2j - k, v = 5i - 3j - 2k **Multiple Choice (Problem 8):** Question: If u and v are orthogonal vectors and u is one of them, whose projection on v is given, which among the given choices represents the result of adding two orthogonal vectors, with one of them being u? (Possible choices are given in the image, but they are missing here)