Write the vector equation of a line given by x-5 3 = y+4 7 = z-6 … - Vidyadip

Question

Write the vector equation of a line given by $\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2}.$

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Answer

We have
$\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2}$
The given line passes through the point (5, -4, 6) and has direction ratios proportional to 3, 7, 2.
Vector equation of the given line passing through the point having position vector $\vec{\text{a}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}$ and parallel to a vector $\vec{\text{b}}=3\hat{\text{i}}+7\hat{\text{j}}+2\hat{\text{k}}$ is
$\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}}$
$\Rightarrow\vec{\text{r}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}+\lambda\big(3\hat{\text{i}}+7\hat{\text{j}}+2\hat{\text{k}}\big)$

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