Question

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 2\hat{k} \).

Hint:

For a line passing through a point and parallel to a vector, use the parametric equations involving the direction ratios.

Answer 1

Step 1: General equation of a line.
The parametric form of the equation of a line passing through a point \( (x_1, y_1, z_1) \) and parallel to the vector \( \mathbf{v} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by: \[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}. \]

Step 2: Apply the given data.
The line passes through the point \( (1, 2, 3) \) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 2\hat{k} \). Thus, the direction ratios are \( a = 3 \), \( b = 2 \), and \( c = -2 \). The equation of the line becomes: \[ \frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z - 3}{-2}. \]

Step 3: Conclusion.
The parametric equations for the line are: \[ x = 1 + 3t, y = 2 + 2t, z = 3 - 2t, \] where \( t \) is the parameter.