1. Draw a second quadrant vector B. (remember that boldface characters represent vector quantities). Show the standard angle [31 for this vector (= angle that B makes with the positive x- axis). Also show the angle that B makes with the positive yaxis: call the latter angle [32. a) Prove the following formulas for the components of B involving the standard angle (hint: start with the formulas for the components based on the angle [32 and then use co-function identities linking cosine and sine of B2 to cosine and sine of [3,) - B,=BcosBl B,=BsinBl b) Prove the following formulas for the components of B: B,=Bcos[32 B,=Bcos|32 c) Draw a right-handed 3D Cartesian coordinate system (= x, y and z axes). Show a vector A with tail in the origina and sticking out in the positive x, y and 2 directions. Show the angles between A and the positive x, y and z axes, and call these angles 01,, uzand a, Prove that A = Acos a, i + Acos a, j + Acos a, k Lesson to be learned: the standard angle approach works well in ZD and relates nicely to the connection between cosine and sine with x-coordinate and ycoordinate, respectively, as explained in the lectures. Unfortunately, this connection cannot be extended to 30! The angles that a vector makes with the positive coordinate axes are called the directional angles, not standard angles ([31 and B, for the 2D vector B, and (1,, a, and a, for the 3D vector A). In both 2D and 3D the components can be calculated using the cosine of these angles, aka the directional cosines. 2. In the lectures, it was proven that: The proof consisted of solving the velocity-versus-time equation for first, and substituting that in the position-versus-time equation next. Show that you can obtain the same result by the reverse procedure: solve the position-versustime equation for first, and then substitute that in the velocity-versus-time equation