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Question
Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line `vec("r") = (-2hat"i"+3hat"j") +lambda(2hat"i"-3hat"j"+6hat"k").`Also, find the distance between these two lines.
Solution
It is given that line passes through the point (2,3,2) and is parallel to the line
`vec("r")=(2hat"i"+3hat"j")+ lambda (2hat"i"-3hat"j" +6hat"k").`
i.e. required line is parallel to the vector `2hat"i" -3hat"j" +6hat"k".`
Equation of the required line is `vec("r")= (2hat"i" + 3hat "j"+2hat"k") +lambda(2hat"i"-3hat"j"+6hat"k")`
The two lines are parallel, we have
`vec("a"_1)=2hat"i"+3hat"j",vec("a"_2)=2hat"i"+3hat"j"+2hat"k"`and`vec("b")=2hat"i"-3hat"j"+6hat"k"`
Therefore, the distance between the lines is given by
`"d"=|(vec("b")xx(vec("a"_2)-vec("a"_1)))/|vec("b")||= ||(hat"i",hat"j",hat"k"),(2,-3,6),(4,0,2)|/sqrt(4+9+36)|`
`=|(-6hat"i"+20hat"j"+12hat"k")/sqrt49|=sqrt580/sqrt49=(2sqrt145)/7`
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