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...S solutions to exercises vectors and matrices a dir b c i j k let p be its tail q head then oq op therefore ox px pq sop roq replace by r in above use the condition is not redundant since there are two of o length making an angle with wind w v bi aj elementary trigonometry cos etc if t u direction cosines some ti uj unit vector so conversely this relation holds letting sides third side line joining midpoints which parallel half d four suitably oriented we have from midpoint similarly other thus opposite equal shows gure parallelogram vertices x on pr y or oy dot product orthogonal acute e place cube rst octant origin at one corner three edges longest diagonal face...