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Question
Find the vector equation of a line passing through the point `hati + 2hatj + 3hatk` and perpendicular to the vectors `hati + hatj + hatk` and `2hati - hatj + hatk`.
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Solution
Let `barb = hati + hatj + hatk` and `barc = 2hati - hatj + hatk`
The vector perpendicular to the vectors `barb` and `barc`, is given by
`barb xx barc = |(hati, hatj, hatk),(1, 1, 1),(2, -1, 1)|`
= `hati(1 + 1) - hatj(1 - 2) + hatk(-1 - 2)`
= `2hati + hatj - 3hatk`
Since the line is perpendicular to the vector `barb` and `barc`, it is parallel to `barb xx barc`.
The vector equation of the line passing through `A(bara)` and parallel to `barb xx barc` is `barr = bara + λ(barb xx barc)`, where λ is a scalar.
Here, `bara = hati + 2hatj + 3hatk`
Hence, the vector equation of the required line is `barr = (hati + 2hatj + 3hatk) + λ(2hati + hatj - 3hatk)`.
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