Find a direction vector \[ \mathbf{d} = \begin{pmatrix}d_1 \\ d_2 \\d_3 \end{pmatrix}\]for the line through $B = (1, 1, 2)$ and $C= (2, 3, 1)$ such that $d_1 + d_2 + d_3 = 10.$
\( \mathbf{d} = \begin{pmatrix}d_1 \\ d_2 \\d_3 \end{pmatrix}\)
\(for \ the \ line \ through \ B = (1, 1, 2)\ and \ C= (2, 3, 1) \ such \ that \ d_1 + d_2 + d_3 = 10.\)
A direction vector = (2 - 1, 3 - 1, 1 - 2) = ( 1, 2 , -1)
So....the sum of the components of the direction vector = 2
This implies that if we multiply each component by 5, then the sum of the components = 10
So we have 5( 1, 2 , - 1) = (5, 10, -5) = ( d1, d2, d3)
And the sum of d1 + d2 + d3 = 10
The equation of the line can be
→
r = (1,1,2) + t (5, 10, - 5)
If t = (1/5) then we get ( 2, 3, 1) = C