The Cartesian equations of a line are . Find the fixed point through which it passes, its direction ratios and also its vector equation.

Answer

Given: Cartesian equation of line are

Io find: fixed point through which the line passes through, its direction ratios and the vector equation.

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 and also its direction ratio.

Explanation:

The Cartesian form of the line can be rewritten as:

\begin{array}{l}

\frac{\mathrm{x}+\frac{1}{3}}{\frac{1}{3}}=\frac{\mathrm{y}-\frac{1}{3}}{\frac{1}{6}}=\frac{\mathrm{z}-1}{-1}=\lambda \\

\Rightarrow \frac{\mathrm{x}+\frac{1}{3}}{2}=\frac{\mathrm{y}-\frac{1}{3}}{1}=\frac{\mathrm{z}-1}{-6}=\lambda

\end{array}

Therefore, and

So, the line passes through and direction ratios of the line are and vector form is:

-\frac{-2}{1}=\frac{-1}{3} \hat{i}-1 \cdot \frac{1}{3} \hat{j}-+\hat{k}-1-\lambda(2 \hat{i}-1-\hat{\jmath}-\xi \hat{k})