# True or False: The vector equation of the line $\Large \frac{x-5}{3}\;=\frac{y+4}{7}\;=\frac{z-6}{2}$ is$\overrightarrow{r}=(5\hat i-4\hat j+6\hat k)+\lambda(3i\;+7j\;+2k).$

$\begin{array}{1 1} True \\ False \end{array}$

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• Vector equation of a line passing through a point and parallel to a given vector is $\overrightarrow r=\overrightarrow a+\lambda \overrightarrow b$
The given equation of the line $\Large \frac{x-5}{3}\;=\frac{y+4}{7}\;=\frac{z-6}{2}$
This is in the cartesian form.
Where $(5,-4,6)$ are the coordinate of the point A and $(3,7,2)$ are the direction ratios of the parallel vector $\overrightarrow b$
The vector equation of the line passing through a point and parallel to a given vector is $\overrightarrow r=\overrightarrow a+\lambda \overrightarrow b$
Hence the required vector equation of the line
$\overrightarrow{r}=(5\hat i-4\hat j+6\hat k)+\lambda(3i\;+7j\;+2k).$
Hence the statement is $True$