Find The Equation Of The Line In Vector And In Cartesian Form That Passes Through The Point With Position Vector 2i^− J^+4 K^ And Is In The Direction I^+2 J^− K^.

Given:

The line passes through the point with position vector:

[latex] \vec{a} = 2\hat{i} – \hat{j} + 4\hat{k}[/latex]—[1] [latex]\vec{b} = \hat{i} + 2 \hat{j} – \hat{k}[/latex]—[2]

As we know, a line through a point with position vector [latex]\vec{a} [/latex]and parallel to

[latex]\vec{b}[/latex] is given by the equation:

[latex]\vec{r} = \vec{a} + \lambda\vec{b}[/latex] [latex]\Rightarrow \vec{r} = 2\hat{i} – \hat{j} + 4\hat{k} + \lambda (\hat{i} + 2\hat{j} – \hat{k})[/latex]

This is the required equation of the line in vector form.

[latex]\Rightarrow x\hat{i} – y \hat{j} + z\hat{k} + (\lambda + 2) \hat{i} + (2\lambda – 1)\hat{j} + (-\lambda + 4)\hat{k}[/latex]

Now by eliminating λ, we get the Cartesian form equation:

[latex]\frac{x-2}{1} = \frac{y+1}{2} = \frac{z-4}{1}[/latex]

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