By Smith, Charles, 1844-1916

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**Extra resources for An elementary treatise on solid geometry**

**Example text**

1. Find the perpendicular distance of an angular point of a cube from a diagonal which does not pass through that angular point. Ex. 2. a^\. line of intersection of Ex. 3. Shew that the two lines x - 2 = 2y - 6 = 3z, 4x - 11 = 4y - 13 = 3z meet in a point, and that the equation of the plane on which they lie is Ex. 4. H Of *b fi trough the point ^w& z ~ "x througli the line whose equations are a. y Ans. (a , /3 , 7% and P _ ~~ y a - 1 I Ex. 5. The shortest distances between the diagonal and the edges which it does not meet are of a rectangular parallelepiped be where a, 6, c are Ex.

The equation of the 11 plane through three given points. / 2 , , ^2 ), (xs ys ^3 ). , , The general equation Ax -f By 4- Cz + D = 0. If the three given points are on this plane, + By\ + Cfc + Ax^ t Ax + By^ + Gz + D and 9 we have = 0, Z> 0. s D from these four equations, we Eliminating A, B, 0, have for the required equation x , y , x. / , -0. i and 8 = be the equations of two planes, be the general equation of a plane through their intersection. For, since S and S are both of the first and hence S \8 = represents degree, so also is If 8=0 S X S = will 17.

Mm + nn 0. N are M, proportional to the direction-cosines of a line, the actual direction-cosines will be ___ M L N Hence the angle between two lines whose direction-cosines and L N respectively is M, N are proportional to L, * r>r\o 1 y( The condition 7 , LL + MM + NN + ^ + ^ *j (L* + M* + N*y 1 2 M } . 2 a ) of perpendicularity is as before LL + MM + NN = 0. Ex. 1. Ex. 2. Shew Shew that the 3x= -y= Ex. that the lines ~ line = = * | and ^ 4z=3t/= -Z = is - are -^ 1 at right angles. perpendicular to the line -4z.