Find the equations of the line passing through the point and parallel to the line . Also find the vector form of this equation so obtained.

Answer

Given: line passes through and is parallel to the line

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

To find: equation of line in vector and Cartesian form

Formula Used: Equation of a line is

Vector form:

Cartesian form:

where is a point on the line and is a vector parallel to the line.

Explanation:

Since the line say is parallel to another line say has the same direction ratios as that of

Here,

Since the equation of is

\begin{array}{l}

\frac{x-6}{3}=\frac{y-2}{-1}=\frac{z+7}{5} \\

\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-4 \hat{\mathrm{l}}+5 \hat{\mathrm{k}}

\end{array}

Therefore,

Vector form of the line is:

\vec{r}=\hat{\imath}-2 \hat{\jmath}+3 \hat{k}-\lambda-\lambda(3 \hat{i}-4 \hat{\jmath}+\sqrt{5} \hat{k})

Cartesian form of the line is:

$\frac{x\u20131}{3}=\frac{y+2}{\u20134}=\frac{z\u20133}{5}$\frac{x-1}{3}=\frac{y+2}{-4}=\frac{z-3}{5}