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Question
Obtain the vector equation of the line `(x + 5)/(3) = (y + 4)/(5)= (z + 5)/(6)`.
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Solution
The cartesian equations of the line are `(x + 5)/(3) = (y + 4)/(5)= (z + 5)/(6)`.
This line is passing through the point A(– 5, – 4, – 5) and having direction ratios 3, 5, 6.
Let `bar"a"` be the position vector of the point A w.r.t. the origin and `bar"b"` be the vector parallel to the line.
Then `bar"a" = -5hat"i" - 4hat"j" - 5hat"k" and bar"b" = 3hat"i" + 5hat"j" + 6hat"k"`.
The vector equation of the line passing through `"A"(bara)` and parallel to `bar"b" "is" bar"r" = bar"a" + lambdabar"b"` where λ is a scalar.
∴ the vector equation of the required line is
`bar"r" = (-5hat"i" - 4hat"j" - 5hat"k") + lambda(3hat"i" + 5hat"j" + 6hat"k")`.
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