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Find the distance between \(Q = (3, -7, -1)\) and the line through \(A = (1, 1, 2)\) and \(B = (2, 3, 4).\) This distance is equal to \(\dfrac{\sqrt{d}}{3}\)
for some integer \(d\). What is \(d\)?

 Nov 8, 2019
 #1
avatar
+1

The distance is sqrt(30)/3, so d = 30.

 Nov 9, 2019
 #2
avatar+2547 
+1

How did you get 30?
 

What are the steps involved?

CalculatorUser  Nov 9, 2019
 #3
avatar+106539 
+1

We can find the  direction vector, AB  =  s ,  of the line thusly

 

( 2 - 1, 3 - 1, 4 - 2)   =  ( 1, 2,  2)  = s

 

Next....find the vector  QA  =  ( 1 - 3, 1  - - 7, 2 - - 1)  =  ( -2, 8, 3)

 

Next.....form the cross product  of    QA  x s  =

 

i      j       k       i      j

-2   8      3      - 2   8     =

1    2      2      1     2

 

[ i * 8 * 2  + j * 3 * 1 + k * -2 * 2]  -  [ k * 8 * 1 + i * 3 * 2  + j * - 2 * 2 ]  =

 

[ 16i  + 3j - 4k] - [ 8k + 6i - 4j]  =

 

10i + 7j  - 12 k   

 

The distance   =

 

Length  of   [QA x s]

_________________   =

Length  of   [ s ]

 

 

√ [ 10^2 + 7^2 + (-12)^2  ]                 √ 293         √ 293

_______________________   =     _______  =  ______  

√ [ 1^2 + 2^2 + 2^2 ]                           √9                 3

 

So  d  = 293

 

 

cool cool cool

 Nov 9, 2019

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