Question

Find the cartesian equation of the line that passes through the point with position vector \( \hat{i}+2\hat{j}+\hat{k}\) and is in the direction of the vector \( \hat{i}-2\hat{j}+3\hat{k}\) ?

1. \(\frac{x-1}{1}=\frac{y-2}{-2}=\frac{z-1}{3}\)
2. \(\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-1}{3}\)
3. \(\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-1}{3}\)
4. \(\frac{x-1}{1}=\frac{y+2}{2}=\frac{z-3}{1}\)

Answer 1

Correct Answer - Option 1 : \(\frac{x-1}{1}=\frac{y-2}{-2}=\frac{z-1}{3}\)

Concept:

The cartesian equation of line through a point (x₁, y₁, z₁) and having direction ratios a, b, c is given by  \(\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}\) .

Calculation:

As it is given that, the required line passes through the point with position vector \( \hat{i}+2\hat{j}+\hat{k}\) 

⇒ x₁ = 1, y₁ = 2, z₁ = 1.

As it is given that, the required line is in the direction of the vector \( \hat{i}-2\hat{j}+3\hat{k}\) i.e the required line is parallel to the vector \( \hat{i}-2\hat{j}+3\hat{k}\)

⇒ a = 1, b = - 2, c = 3.

∴The Cartesian equation of the required line is: \(\frac{x-1}{1}=\frac{y-2}{-2}=\frac{z-1}{3}\)

Hence, option 1 is correct.