(Solved): Problem 2: You have two vectors: \( \boldsymbol{A}=2 \mathbf{i}-1 \mathbf{j}+4 \mathbf{k} \) and \( ...

Problem 2: You have two vectors: \( \boldsymbol{A}=2 \mathbf{i}-1 \mathbf{j}+4 \mathbf{k} \) and \( \boldsymbol{B}=-4 \mathbf{i}+2 \mathbf{j}-3 \mathbf{k} \). Answer the following questions based on this information. a. Use the dot product to determine the projection of \( \boldsymbol{A} \) along \( \boldsymbol{B} \) ? Remember, in order to compute projections you need to use a unit vector. Therefore, you must decide...do we need the unit vector for \( \boldsymbol{A} \) or for \( \boldsymbol{B} \) ? b. Use the dot product to determine the angle \( \theta \) between the two vectors \( \boldsymbol{A} \) and \( \boldsymbol{B} \). c. Confirm your previous answers by computing the projection of \( \boldsymbol{A} \) along \( \boldsymbol{B} \) given by \( |\mathrm{A}| \cos \theta \). d. In general, what can you say about two vectors if the dot product is zero? e. Let's talk about units. Notice that no units were given for this problem. Consider the following scenarios: i. Suppose \( \mathbf{A} \) was a force vector with units of " \( N \) " and \( \mathbf{B} \) was a distance vector with units of " m ." What would be the units of the \( \operatorname{dot} \) product \( \mathbf{A} \cdot \mathbf{B} \) ? ii. The dot product can also be used in physics to express the quantity called "work." Do the units noted above agree with your recollection for the units of "work?" iii. Suppose \( \mathbf{A} \) was a distance vector with units of " ft " and \( \mathbf{B} \) was a distance vector with units of " ft ." What would be the units of the \( \operatorname{dot} \) product \( \mathbf{A} \cdot \mathbf{B} \) ? iv. Are the units noted above accurate for measuring the projection of \( \mathbf{A} \) along \( \mathbf{B} \) ? v. Suppose \( \mathbf{A} \) was a force vector with units of "lb" and \( \mathbf{B} \) was a unit vector (unit-less). What would be the units of the \( \operatorname{dot} \) product \( \mathbf{A} \cdot \mathbf{B} \) ? vi. Are the units noted above accurate for measuring the projection of \( \mathbf{A} \) along \( \mathbf{B} \) ? Moral of the story: If you want to compute the projection of one vector along another vector, you must express the direction of interest as a unit vector.