Find the vector equation of a line which is parallel to the vector `2 hat i- hat j+3 hat k` and which passes through the point (5, -2,4). Also reduce it to Cartesian form.

Explanation

Vector equation of the given line:


The given line passes through the point `A(5,-2,4)` and it is parallel to the vector `vec{m}=(2 hat{i}-hat{j}+3 hat{k})`


the position vector of `{A}` is given by `vec{r}_{1}=(5 hat{i}-2 hat{j}+4 hat{k})`


Hence the vector equations of the given line is


`vec{r}=vec{r}_{1}+lambda vec{m}`


` Rightarrow vec{r}=(5 hat{i}-2 hat{j}+4 hat{k})+lambda(2 hat{i}-hat{j}+3 hat{k}) quad ldots . .(i)`


Cartesisan equation of the given line:


Taking `vec{r}=(x hat{i}+y hat{j}+z hat{k})` equation (i) becomes :


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