Find the vector and the cartesian equations of a line that passes through the point $A(1, 2, -1)$ and parallel to the line $5x - 25 = 14 - 7y = 35z$.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $\vec{r} = (\hat{i} + 2\hat{j} - \hat{k}) + \lambda(7\hat{i} - 5\hat{j} + \hat{k})$; $\frac{x-1}{7} = \frac{y-2}{-5} = \frac{z+1}{1}$ ##
Given point $A(1, 2, -1)$
Given line $5x - 25 = 14 - 7y = 35z$,
$\Rightarrow 5(x - 5) = -7(y - 2) = 35z$,
$\Rightarrow \frac{x-5}{7} = \frac{y-7}{-5} = \frac{z}{1}$ [Divide by 35]
Direction ratio of the line $(7, -5, 1)$
$∴$ Direction ratio of the parallel line $(7, -5, 1)$
Equation of the line passing through the point $A(1, 2, -1)$ and parallel to the given line
$\frac{x-1}{7} = \frac{y-2}{-5} = \frac{z+1}{1}$
Vector form of the line
$\hat{i} + 2\hat{j} - \hat{k} + \lambda(7\hat{i} - 5\hat{j} + \hat{k})$