Question:

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

Updated On: Sep 20, 2023
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Solution and Explanation

It is given that the line passes through the point with the position vector
\(\vec a\)=2\(\hat i\)-\(\hat j\)+4\(\hat k\)...(i)
\(\vec b\)=\(\hat i\)+2\(\hat j\)-\(\hat k\)....(ii)

It is known that a line through a point with position vector a→ and parallel to b→ is given by the equation,
\(\vec r\)=\(\vec a\)\(\vec b\)
⇒ \(\vec r\)=2\(\hat i\)-\(\hat j\)+4\(\hat k\)+λ(\(\hat i\)+2\(\hat j\)-\(\hat k\))

This is the required equation of the line in vector form.
r→=x\(\hat i\)-y\(\hat j\)+z\(\hat k\)
⇒x\(\hat i\)-y\(\hat j\)+z\(\hat k\)=(λ+2\(\hat i\)+(2λ-1)\(\hat j\)+(-λ+4)\(\hat k\)

Eliminating, we obtain the cartesian form equation as \(\frac{x-2}{1}\)=\(\frac{y+1}{2}\)=\(\frac{z-4}{-1}\)

This is the required equation of the given line in cartesian form.

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Concepts Used:

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Vector Equation

Consider a line that passes through a given point, say ‘A’, and the line is parallel to a given vector '\(\vec{b}\)‘. Here, the line ’l' is given to pass through ‘A’, whose position vector is given by '\(\vec{a}\)‘.  Now, consider another arbitrary point ’P' on the given line, where the position vector of 'P' is given by '\(\vec{r}\)'.

\(\vec{AP}\)=𝜆\(\vec{b}\)

Also, we can write vector AP in the following manner:

\(\vec{AP}\)=\(\vec{OP}\)\(\vec{OA}\)

𝜆\(\vec{b}\) =\(\vec{r}\)\(\vec{a}\)

\(\vec{a}\)=\(\vec{a}\)+𝜆\(\vec{b}\)

\(\vec{b}\)=𝑏1\(\hat{i}\)+𝑏2\(\hat{j}\) +𝑏3\(\hat{k}\)