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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 $d_1 + d_2 + d_3 = 10.$

 Jul 30, 2019
 #1
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\( \mathbf{d} = \begin{pmatrix}d_1 \\ d_2 \\d_3 \end{pmatrix}\)

 

\(for \ the \ line \ through \ B = (1, 1, 2)\ and \ C= (2, 3, 1) \ such \ that \ d_1 + d_2 + d_3 = 10.\)

 

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

 

cool cool cool

 Jul 30, 2019

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