By the end of Week 2, you should: · Be familiar with 2D and 3D Cartesian coordinate systems · 3D: Know direction cosine representation of a vector . 3D: Be able to translate back and forth between the Cartesian representation of a vector, and its representation via direction cosines · Read 3D diagrams · Know (and be able to apply) the following projection techniques: o Using trigonometric functions of given angles o Using "slope triangles" o "Two-step projection technique" · Define a unit vector using the direction cosines · Know the relationship between the direction cosines of a vector . Find the resultant force given the forces acting on the body · Find parameters of certain forces when some other forces and the resultant force are given · Understand the difference between a position vector and a displacement vector . Be able to write a displacement vector in Cartesian components knowing the coordinates of its head and tail . Be able to express in Cartesian components a force, if you know the coordinates of two points on its line if action · Apply it to solving problems with forces acting along ropes, struts, cables ... . Optional (though recommended): understand the "calculational trick" for the force defined by two points on its line of action, and understand why we think this "trick" is convenient . Using your calculator, be able to solve a linear system of up to 6 equations in 6 unknowns · Know two definitions (the one with the cosine and absolute values, the other with Cartesian components) of a dot product of two vectors . Be able to use these two definitions to determine the angle between two vectors . Understand the connection between the dot product of two vectors and projection of one of the two vectors on the line of action of the other vector · Be able to use the dot product to calculate the projection of one vector onto another Questions . Assume that you have one vector equation: R = A +B +C. How many scalar equations does this equation correspond to in 3D? · Why and when do we need to use the "two-step projection technique"? . Let A = 5 1+ 3 ] - 2 k . Write down a unit vector in the direction of A. . Let B = 31+ 5 ] + 2 k, and A is defined above. Find the angle between A and B . . For arbitrary A and B , is the projection of B on A equal to the projection of A on B ? . A force of 100 N acts along the line passing through A(5,3,-2) and B(4,3,2), in the direction from A to B. Write the expression for this force in Cartesian components.