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Straight line in space — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsStraight line in space3 Marks
Question
A line passes throuth the point with position vector $2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$ and is in the direction of $3\hat{\text{i}}+4\hat{\text{j}}-5\hat{\text{k}}.$ Find equations of the line in vector and cartesian form.
✓

Answer

We know that the vector equation of a line passing through a point with position vector $\vec{\text{a}}$ and parallel to the vector $\vec{\text{b}}$ is $\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}}.$
Here,
$\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$
$\vec{\text{b}}=3\hat{\text{i}}+4\hat{\text{j}}-5\hat{\text{k}}$
So, the vector equation of the required line is
$\vec{\text{r}}=\big(2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}\big)+\lambda\big(3\hat{\text{i}}+4\hat{\text{j}}-5\hat{\text{k}}\big)$
Here, $\lambda$ is a parameter.

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