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The distance of the point (-hati+2hatj+6...

The distance of the point `(-hati+2hatj+6hatk)` from the straight line through the point `(2,3,-4)` and parallel to `6hati+3hatj-4hatk` is

A

`2sqrt(3)`

B

`4sqrt(3)`

C

6

D

7

Text Solution

Verified by Experts

The correct Answer is:
D
Let point P whose position vector is `(-hati+2hatj+6hatk)` and a stright line passing through Q(2, 3, -4) parallel to the vector `n=6hati+3hatj-4hatk`.

`because` Required distance d=Projection of line segment PQ perpendicular to vector n.
`=(|PQxxn|)/(|n|)`
Now, `PQ=3hati+hatj-10hatk`, so
`PQxxn=|{:(hati," "hatj," "hatk),(3," "1,-10),(6," "3,-4):}|=26hati-48hatj+3hatk`
So, `d=(sqrt((26)^(2)+(48)^(2)+(3)^(2)))/(sqrt((6)^(2)+(3)^(2)+(4)^(2)))`

`=sqrt((676+2304+9)/(36+9+16))=sqrt((2989)/(61))`
`=sqrt(49)=7`units
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