Determine the value of k that makes the lines `(x+2)/4`=`(y+1)/5`=`(z-3)/3` and `vecr`=(1,3,6) +s(-2k,2,k), seR, perpendicular. a) 1 b) 3 c) -3 d) none of the above

Expert Answers

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You need to identify the direction vectors to each line, hence, the direction vector of the line given by symmetric equations is `barv_1 = lt4,5,3gt`  and the direction vector of the line given by parametric equations is `barv_2=lt-2k,2,kgt.`

The problem provides the information that the lines are orthogonal, hence, the cross product of direction vectors needs to be zero such  that:

`barv_1*barv_2=0`

`barv_1*barv_2=lt4,5,3gt*lt-2k,2,kgt`

`lt4,5,3gt*lt-2k,2,kgt = 4*(-2k) + 5*2 + 3*(k)`

`-8k + 10 + 3k = 0 =gt -5k = -10 =gt k = (-10)/(-5)`

`k = 2`

Hence, the lines are orthogonal for k=2, thus the option d) (none of the above) represents the answer.

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