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  • The cartesian equation of a line is x - 5/3 = y+ 4/ 7 = z- 6/ 7.Write its vector form.

7.  The cartesian equation of a line is \frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{7}.  Write its vector form.

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Given the Cartesian equation of the line;

\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{7}

Here the given line is passing through the point (5,-4,6). 

So, we can write the position vector of this point as;

\vec{a} = 5\widehat{i}-4\widehat{j}+6\widehat{k}

And the direction ratios of the line are 3, 7, and 2.

This implies that the given line is in the direction of the vector, \vec{b} = 3\widehat{i}+7\widehat{j}+2\widehat{k}.

Now, we can easily find the required equation of line:

As we know that the line passing through the position vector \vec{a} and in the direction of the vector \vec{b} is given by the relation,

\vec{r} = \vec{a} + \lambda \vec{b},\ \lambda \epsilon R

So, we get the equation.

\vec{r} = 5\widehat{i}-4\widehat{j}+6\widehat{k} + \lambda(3\widehat{i}+7\widehat{j}+2\widehat{k}),\ \lambda \epsilon R

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

 

Posted by

Divya Prakash Singh

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