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By Smith, Charles, 1844-1916

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14] that the direction-cosines of the normal to the plane are proportional to A, B, C. Hence by Article 24 the angle between the planes whose equations are + IS < V (A* Cz AA +Bff+Oa B C V (A 4- JT 4- C * 2 2 4- +Z=0 4- ) 2 ) Ex. 1. Find the equation of the plane containing the line 2x + By + 4z = 5, and perpendicular to the plane x - y + z = 0. Ans. Ex. 1 2. At what angle do the planes x + y + z = 4, origin in the acute angle or in the obtuse? acute angle or in the obtuse ? x- x-2y-z= cut ?

Of XOY Let X, fi, v be the angles YOZ ZOX, respectively. direction-ratios of the line PQ, and let I, m, n be the Let PL, LK, a, /Q, 7 be the angles it makes with the axes. KQ be parallel to the axes so that PL = l. PQ, as in Art. 39. Then, since the projection of PQ on the axis of x is equal to the projection of PLKQ, we have , Let . PQ cos a = PL + LKcos v + KQ cos ^ n cos cos a = I + m cos v = I cos v + m + n cos X, cos /3 therefore -f- Similarly and cos 7 To find 41. = I cos /z +m cos X ; /i.

To change 45. the direction of the axes without changing the origin, both systems being rectangular. m m Let l v v n^ /2 ra n and lv y n s be the direction? cosines ines of the new axes referred to the old. , , \ <2 Let P be any point are x, y, z and x, Draw PL pendicular to y , z whose co-ordinates in the two systems . X OY and LM per ML and LP=z OX is equal to the sum of perpendicular to the plane = x, =y then ; OM OX Since the projection of OP on the projections of OM, and LP, ML Similarly y= and z m = we have a + ra 2 y + mn z, x + n2 y + n s z l .

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An elementary treatise on solid geometry by Smith, Charles, 1844-1916


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