Page 179 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 179
170 VECTORS [CHAP. 7
Fig. 7-29
or ðr r 1 Þ ðr 2 r 1 Þ ðr 3 r 1 Þ¼ 0
In terms of rectangular coordinates this becomes
½ðx x 1 Þi þðy y 1 Þj þðz z 1 Þk ½ðx 2 x 1 Þi þðy 2 y 1 Þj þðz 2 z 1 Þk
½ðx 3 x 1 Þi þðy 3 y 1 Þj þðz 3 z 1 Þk¼ 0
x x 1 y y 1
z z 1
z 2 z 1 ¼ 0
x 2 x 1 y 2 y 1
or, using Problem 7.24,
x 3 x 1 y 3 y 1 z 3 z 1
7.28. Find an equation for the plane passing through the points P 1 ð3; 1; 2Þ, P 2 ð 1; 2; 4Þ, P 3 ð2; 1; 1Þ.
The positions vectors of P 1 ; P 2 ; P 3 and any point Pðx; y; zÞ on the plane are respectively
r 1 ¼ 3i þ j 2k; r 2 ¼ i þ 2j þ 4k; r 3 ¼ 2i j þ k; r ¼ xi þ jj þ zk
Then PP 1 ¼ r r 1 , P 2 P 1 ¼ r 2 r 1 , P 3 P 1 ¼ r 3 r 1 ,all lie in the required plane and so the required
equation is ðr r 1 Þ ðr 2 r 1 Þ ðr 3 r 1 Þ¼ 0, i.e.,
fðx 3Þi þðy 1Þj þðz þ 2Þkg f 4i þ j þ 6kg f i 2j þ 3kg¼ 0
fðx 3Þi þðy 1Þj þðz þ 2Þkg f15i þ 6j þ 9kg¼ 0
15ðx 3Þþ 6ðy 1Þþ 9ðz þ 2Þ¼ 0 or 5x 2y þ 3z ¼ 11
Another method: By Problem 7.27, the required equation is
x 3 y 1
z þ 2
1 3 2 1 4 þ 2 ¼ 0 or 5x þ 2y þ 3z ¼ 11
2 3 1 11 þ 2
