Finding coordinates on line closest to point

[TsAmE]

Let l1 be the line passing through the point A(5,-6,7) parallel to the x-axis. Write down the coordinates of the point B on l1 which is closest to C(1,3,4).

I dont know where to start. I tried searching on internet for some info on this section but found nothing

 

[soroban]

Hello, TsAmE!

You didn't make a sketch, did you?

\(\displaystyle \text{Let }L\text{ be the line passing through the point }A(5,\text{-}6,7)\text{ parallel to the }x\text{-axis.}\)

\(\displaystyle \text{Write down the coordinates of the point }B\text{ on }L\text{ which is closest to }C(1,3,4).\)

      |
      |   C
      |   * (1,3.4)
      |
------+------------------
      |
      |
      |         A
. - - + - - - - * - - L
      |       (5,-6.7)
      |

Can you point to where you think \(\displaystyle B\) might be?

 

[TsAmE]

Oh sorry please refer to the attachment for my sketch.

Would the closest distance between B and C be when the line BC is perpendicular to l1? If so how would you then find the coordinates of B?

 

[galactus]

Since the given line with the point (5,-6,7) is parallel to the x-axis, it would appear the needed point will have the same y and z coordinates:

y=-6 and z=7

Since the point is at a right angle to (1,3,4), do you see what x should be?.

"the line is parallel to the x-axis" is the clincher clue. No fancy-schmancy calculations needed. Just observation.

 

[TsAmE]

I understand now, thanks a lot