Question

Find the cartesian and vector equations of the line passing through \( A(1, 2, 3) \) and having direction ratios \( 2, 3, 7 \).

Hint:

To find the cartesian equation of the line, express the parametric equations in terms of \( t \), then eliminate \( t \).

Answer 1

Step 1: Vector equation of the line.
The vector equation of the line is given by: \[ \vec{r} = \vec{r_0} + t \cdot \vec{d} \] Where \( \vec{r_0} \) is the position vector of the point \( A(1, 2, 3) \), and \( \vec{d} = \langle 2, 3, 7 \rangle \) is the direction vector. Thus, the vector equation of the line is: \[ \vec{r} = \langle 1, 2, 3 \rangle + t \cdot \langle 2, 3, 7 \rangle \] Which simplifies to: \[ \vec{r} = \langle 1 + 2t, 2 + 3t, 3 + 7t \rangle \]

Step 2: Cartesian equation of the line.
To obtain the cartesian equation, solve for \( t \) from each of the parametric equations: \[ \frac{x - 1}{2} = t, \frac{y - 2}{3} = t, \frac{z - 3}{7} = t \] Equating the values of \( t \), we get the cartesian equation: \[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{7} \]

Final Answer: The vector equation of the line is: \[ \vec{r} = \langle 1 + 2t, 2 + 3t, 3 + 7t \rangle \] The cartesian equation of the line is: \[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{7} \]