. Plane and solid analytic geometry . ^y DIRECTION COSINES 425. Fig. 4 Lgle between Two Directed Lines. Let it be required tollie angle Q between the two directed lines, L^ and Z2,whose direction angles are «!, ySi, yiand ttg, /82, 72- We can assume without loss ofgenerality that L^ and L^ passthrough the origin.^ Take anypoint P: (ao, ?/o, ^o) o^ ^i? so that thedirection from 0 to P is that of Xi,and draw the broken line OMNP,whose directed segments OM, MN, M^NP are, respectively, the coordi-nates 0^0, 2/oj ^0 of P The projectionof this broken line on L2 equals the projection of OP on L^: (1)

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. Plane and solid analytic geometry . ^y DIRECTION COSINES 425. Fig. 4 Lgle between Two Directed Lines. Let it be required tollie angle Q between the two directed lines, L^ and Z2,whose direction angles are «!, ySi, yiand ttg, /82, 72- We can assume without loss ofgenerality that L^ and L^ passthrough the origin.^ Take anypoint P: (ao, ?/o, ^o) o^ ^i? so that thedirection from 0 to P is that of Xi,and draw the broken line OMNP,whose directed segments OM, MN, M^NP are, respectively, the coordi-nates 0^0, 2/oj ^0 of P The projectionof this broken line on L2 equals the projection of OP on L^: (1) Proj .,^0P= Proj .r^^OM^ Proj .^^ MN + Proj .^^ ^ Ch. Xyil, § 4, ^ 0P= OP , Proj.^^ 0M= \0M\ cos ^ /rOxlf, where K is a point on L2, such that OA has the direction ofL2. I£ the directed line-segment 031 has the direction of thepositive axis of x, as is the case in the figure, we have \0M\ = OM, ^ IWM= a., and therefore, Proj.^^ 0M= OM cos Oo- If OM has the direction of the negative axis of x, 10M\ = - OM and ^ KOM= 180° - ag; in t

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