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Question
The cartesian equations of a line are x = ay + b, z = cy + d. Find its direction ratios and reduce it to vector form.
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Solution
The cartesian equation of the given line is \[x = ay + b, z = cy + d\]
It can be re-written as
\[\frac{x - b}{a} = \frac{y - 0}{1} = \frac{z - d}{c}\]
Thus, the given line passes through the point (b,0,d) and its direction ratios are proportional to a, 1, c. It is also parallel to the vector \[\vec{b} = a \hat{i} + \hat{j} + c \hat{ k} \]
We know that the vector equation of a line passing through a point with position vector ` vec a` and parallel to the vector `vec b` is \[\vec{r} = \vec{a} + \lambda \vec{b}\]
Vector equation of the required line is
\[\vec{r} = \left( b \hat{i} + 0 \hat{j} + d \hat{k} \right) + \lambda \left( a \hat{i} + \hat{j} + c \hat{k} \right)\]
\[\text{ Here }, \lambda \text{ is a parameter } . \]
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