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Question
Find the vector equation of a line passing through a point with position vector `2hati - hatj + hatk` and parallel to the line joining the points `-hati + 4hatj + hatk` and `-hati + 2hatj + 2hatk`.
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Solution
Let A, B and C be the points with position vectors `2hati - hatj + hatk`, `-hati + 4hatj + hatk` and `hati + 2hatj + 2hatk`, respectively.
We have to find the equation of a line passing through the point A and parallel to vector BC.
Now, `vec(BC)` = Position vector of C – Position vector of `vecB`
= `(hati + 2hatj + 2hatk) - (-hati + 4hatj + hatk)`
= `2hati - 2hatj + hatk`
We know that, the equation of a line passing through a position vector `veca` and parallel to vector `vecb` is `vecr = veca + λvecb`
∴ `vecr = (2hati - hatj + hatk) + λ(2hati - 2hatj + hatk)` is the required equation of line in vector from.
[Here, `vec(BC) = vecb`]
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