Question

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

Answer 1

We have

$\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$

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 $\overrightarrow{a}=5\hat{i}-4\hat{j}+6\hat{k}$ and parallel to a vector $\overrightarrow{b}=3\hat{i}+7\hat{j}+2\hat{k}$ is

$$\begin{gathered} \overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b} \\ \Rightarrow \overrightarrow{r}=5\hat{i}-4\hat{j}+6\hat{k}+\lambda \left(3\hat{i}+7\hat{j}+2\hat{k}\right) \end{gathered}$$

Disclaimer: The answer given in the book is incorrect. This solution is created according to the question given in the book.