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Straight line in space — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsStraight line in space5 Marks
Question
Find the vector equation for the line which passes through the point (1, 2, 3) and parallel to the vector $\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}}.$ Reduce the corresponding equation in cartesian form.
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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,
$\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}$
$\vec{\text{b}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}}$
vector equation of the required line is
$\vec{\text{r}}=\big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\big)$
Here, $\lambda$ is a parameter.
Reducing (1) to cartesian form, we get
$\text{x}\hat{\text{i}}+\text{y}\hat{\text{i}}+\text{z}\hat{\text{k}}=\big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}}\big)$ $\big[\text{putting }\vec{\text{r}}=\text{x}\hat{\text{i}}+\text{y}\hat{\text{j}}+\text{z}\hat{\text{k}} \text{ in }(1)\big]$
$\Rightarrow\text{x}\hat{\text{i}}+\text{y}\hat{\text{j}}+\text{z}\hat{\text{k}}=(1+\lambda)\hat{\text{i}}+(2-2\lambda)\hat{\text{j}}+(3+3\lambda)\hat{\text{k}}$
Comparing the cofficients of $\hat{\text{i}},\hat{\text{j}}$ and $\hat{\text{k}},$ we get
$\text{x}=1+\lambda,\text{y}=2-2\lambda,\text{z}=3+3\lambda$
$\Rightarrow\text{x}-1=\lambda,\frac{\text{y}-2}{-2}=\lambda,\frac{\text{z}-3}{3}=\lambda$
$\Rightarrow\frac{\text{x}-1}{1}=\frac{\text{y}-2}{-2}=\frac{\text{z}-3}{3}=\lambda$
Hence, the cartesian form of (1) is
$\frac{\text{x}-1}{1}=\frac{\text{y}-2}{-2}=\frac{\text{z}-3}{3}$

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