The Cartesian equation of a line is 2x - 2 = | vedclass

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Question

(A) The given Cartesian equation is $2x - 2 = 3y + 1 = 6z - 2$. Divide the entire equation by the least common multiple of the coefficients of $x, y,

Vedclass · Accepted Answer

$\bar{r} = \left(\hat{i} - \frac{1}{3}\hat{j} + \frac{1}{3}\hat{k}\right) + \lambda(3\hat{i} + 2\hat{j} + \hat{k})$

Vedclass · Answer

(A) The given Cartesian equation is $2x - 2 = 3y + 1 = 6z - 2$. Divide the entire equation by the least common multiple of the coefficients of $x, y, z$,which is $6$: $\frac{2(x - 1)}{6} = \frac{3(y + 1/3)}{6} = \frac{6(z - 1/3)}{6}$ This simplifies to: $\frac{x - 1}{3} = \frac{y + 1/3}{2} = \frac{z - 1/3}{1}$ The line passes through the point $(1, -1/3, 1/3)$ and has direction ratios $(3, 2, 1)$. The vector equation of a line passing through point $\vec{a}$ with direction vector $\vec{b}$ is $\vec{r} = \vec{a} + \lambda\vec{b}$. Substituting the values,we get: $\vec{r} = \left(\hat{i} - \frac{1}{3}\hat{j} + \frac{1}{3}\hat{k}\right) + \lambda(3\hat{i} + 2\hat{j} + \hat{k})$.

The Cartesian equation of a line is $2x - 2 = 3y + 1 = 6z - 2$. The vector equation of the line is:

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