Question

Find the non-parametric form of vector equation and Cartesian equations of the straight line passing through the point with position vector ${\displaystyle 4\hat{\text{i}}+3\hat{\text{j}}-7\hat{\text{k}}}$ and parallel to the vector ${\displaystyle 2\hat{\text{i}}-6\hat{\text{j}}+7\hat{\text{k}}}$

Answer 1

${\displaystyle \overrightarrow{\text{a}}=4\hat{\text{i}}+3\hat{\text{j}}-7\hat{\text{k}} (x_1,y_1,z_1)}$ = (4, 3, – 7)

${\displaystyle \overrightarrow{\text{b}}=2\hat{\text{i}}-6\hat{\text{j}}+7\hat{\text{k}} (l,\text{m},\text{n})}$ = (2, – 6, 7)

Vector equation

${\displaystyle (\overrightarrow{\text{r}}-\overrightarrow{\text{a}})\times \overrightarrow{\text{b}}}$ = 0

${\displaystyle [\overrightarrow{\text{r}}-(4\hat{\text{i}}+3\hat{\text{j}}-7\hat{\text{k}})]\times (2\hat{\text{i}}-6\hat{\text{j}}+7\hat{\text{k}})}$ = 0

Cartesian equations

${\displaystyle \frac{x-x_1}{l}=\frac{y-y_1}{\text{m}}=\frac{z-z_1}{\text{n}}}$

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