Find the vector and Cartesian equations of the line joining the points whose position vectors are

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Find the vector and Cartesian equations of the line joining the points whose position vectors are

Find the vector and Cartesian equations of the line joining the points whose position vectors are $(\hat{i}-2 \hat{j}+\hat{k})$ and $(\hat{i}+3 \hat{j}-2 \hat{k})$
Answer
Given: line passes through the points whose position vectors are $(\hat{i}-2 \hat{j}+\hat{k})$ and $(\hat{i}+3 \hat{j}-2 \hat{k})$ .
Io find: equation of line in vector and Cartesian forms
Formula Used: Equation of a line is
Vector form: $\overrightarrow{\mathrm{I}}=\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{k}}$
Cartesian form: $\frac{\mathrm{x}-\mathrm{x}_{1}}{\mathrm{~b}_{\mathrm{n}}}=\frac{y-\mathrm{y}_{1}}{\mathrm{~b}_{\mathrm{y}}}=\frac{z-\mathrm{z}_{\mathrm{I}}}{\mathrm{h}_{\mathrm{s}}}=\lambda$ .
where $\vec{a}=x_{1} \hat{l}+y_{1} \hat{l}+z_{1} k$ is a point on the line and $\vec{b}=b_{1} \hat{i}+b_{2} \hat{l}+b_{3} \vec{k}$ with $b_{1}: b_{2}: b_{3}$ being the direction ratios of the line.
Explanation:
Here, $\hat{a}=\hat{i}-2 \hat{i}+\hat{k}$
The direction ratios of the line are $(1-1):(-2-3):(1+2)$