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  • Find the equation of the line in vector and in cartesian form that passes through the point with position vector 2 i caret - j caret + 4 caret and is in the direction i caret + 2 j caret - k caret.

5. Find the equation of the line in vector and in cartesian form that passes through the point with position vector 2\widehat{i}-\widehat{j}+4\widehat{k}and is in the direction  \widehat{i}+2\widehat{j}-\widehat{k}.

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Given that the line is passing through the point with position vector 2\widehat{i}-\widehat{j}+4\widehat{k} and is in the direction of the line \widehat{i}+2\widehat{j}-\widehat{k}.

And we know the equation of the line which passes through the point with the position vector \vec{a}and parallel to the vector \vec{b} is given by the equation,

\vec{r} = \vec{a} +\lambda\vec{b}

\Rightarrow \vec{r} =2\widehat{i}-\widehat{j}+4\widehat{k} + \lambda(\widehat{i}+2\widehat{j}-\widehat{k})

So, this is the required equation of the line in the vector form.

\vec{r} =x\widehat{i}+y\widehat{j}+z\widehat{k} = (\lambda+2)\widehat{i}+(2\lambda-1)\widehat{j}+(-\lambda+4)\widehat{k}

Eliminating \lambda, from the above equation we obtain the equation in the Cartesian form :

\frac{x-2}{1}= \frac{y+1}{2} =\frac{z-4}{-1}

Hence this is the required equation of the line in Cartesian form.

Posted by

Divya Prakash Singh

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