Vector equations of lines Practice

{"questions":[{"content":"Fluency: the first component of B − A for A(4, 2) and B(9, 7) is [[input-vec-t1]]","widgets":{"input-vec-t1":{"type":"input","answer":"5"}},"hints":["Subtract 4 from 9."],"explanation":"The first component is 5."},{"content":"Method recognition: which form describes a line?<br>[[choice-vec-t2]]","widgets":{"choice-vec-t2":{"type":"choice","options":["r = a + λd","r = a · d","r = |a| + |d|","r = a × λ only"],"answer":[0],"explanations":["The scalar parameter moves from a fixed point in the direction d.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["A line needs a point and direction."],"explanation":"The scalar parameter moves from a fixed point in the direction d."},{"content":"Exam execution: directions are scalar multiples. What should be tested next?<br>[[choice-vec-t3]]","widgets":{"choice-vec-t3":{"type":"choice","options":["Whether a point from one line lies on the other","The normal CDF","A base case","A second derivative"],"answer":[0],"explanations":["Point membership determines whether the parallel lines are identical.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Distinguish parallel from coincident."],"explanation":"Point membership determines whether the parallel lines are identical."},{"content":"Synthesis: two parameter equations agree in x and y but disagree in z. Conclude that the lines<br>[[choice-vec-t4]]","widgets":{"choice-vec-t4":{"type":"choice","options":["do not intersect","must intersect","are identical","have zero direction"],"answer":[0],"explanations":["The inconsistent z equation rules out a common point.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["All coordinate equations must agree."],"explanation":"The inconsistent z equation rules out a common point."},{"content":"What should you do next after equating two vector line equations?<br>[[choice-vec-t5]]","widgets":{"choice-vec-t5":{"type":"choice","options":["Solve for the parameters and check every coordinate","Normalize every fixed point","Integrate each vector","Ignore the third coordinate"],"answer":[0],"explanations":["A common point requires a consistent simultaneous solution.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Treat equality component by component."],"explanation":"A common point requires a consistent simultaneous solution."}],"mix":1}