1. Find the volume of the triangular prism with vertices A(1, 3, 4), B(-2, -2, -4) and C(0, 3, -1). (Hint- The volume of a triangular prism is one-half the volume of a parallelepiped). 2. A 4.5 N force which acts along the direction vector (2, 3), moves an object from A(1, 3) to B(3, 7). Find the work done if the given units are metres. 3. Several math students are taking a break from their studies by visiting a playground. One student pushes the others on a merry-go-round. The diameter of the merry-go-round is 2.8 m, and the student pushes with a force of 40 N along the purple vector shown in the diagram. 1100 a. What torque does the student apply? b. At what angle would the student need to apply the force to achieve maximum torque? c. If the student continues to push with a force of 40 N, what maximum torque could be applied?1. a) Explain why it is not possible for a . (b . ) to equal (a . b) . 2 . (This means that the dot product is not associative.) b) Verify using an example that a + (b . ( ) is not equal to (a + b) . (a +2). (This means that addition does not distribute over the dot product.) Explain the problem that arises. 2. Use a specific example to prove that the cross product is also not associative. That is, use three specific vectors in 3-space to show that a *(b * ( ) is not equal to (a x b) x 2. 3. Verify using a specific example that (a + b) * (@ - b) = 2(xa ). Expand to the general case to prove that the result is always true. For help with this question, please check the U1A6 Assignment Question 3 video on YouTube from your instructor. Please note that the proof in the video should show that (a + b) * (a - b) = 2(6*2 ) 4. Use a specific example to explore how the cross product behaves under scalar multiplication. Is it true that k(a * 6) = (ka ) * b = a * (kj )? Expand to the general case to prove your theory. 0 5. Verify (a + 6) * (a + 5) = -. What can be said about two vectors whose cross product is the zero vector? 6. a) Let a = (3, 4, 1), b = (5, -2, 3) and ~ = (0, 1, -3). Find the triple product, a . (b * [). b) Explain why (a . b ) * ~ does not exist