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Solve the following.
The Cartesian equation of a line is: . Find the vector equation of a line passing through and…
The Cartesian equation of a line is: $2x - 3 = 3y + 1 = 5 - 6z$. Find the vector equation of a line passing through $(7, -5, 0)$ and parallel to the given line.
Answer
Answer
AIPut $2x - 3 = 3y + 1 = 5 - 6z = k$: $x = \frac{k+3}{2}$, $y = \frac{k-1}{3}$, $z = \frac{5-k}{6}$.
Direction ratios: $\frac12 : \frac13 : -\frac16 = 3 : 2 : -1$.
The required line passes through $(7,-5,0)$ with direction $3\hat\imath + 2\hat\jmath - \hat k$.
Hence $\vec r = 7\hat\imath - 5\hat\jmath + \lambda(3\hat\imath + 2\hat\jmath - \hat k)$.
Final answer: $\vec{r} = 7\hat{\imath} - 5\hat{\jmath} + \lambda(3\hat{\imath} + 2\hat{\jmath} - \hat{k})$
From ISC 2018 Mathematics Paper 1, question 15(b).
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