Question

Find the Cartesian equation of a line which passes through point (3, -2, -5) and is parallel to the vector \( (3\hat{i} + 2\hat{j} - 2\hat{k}) \).

Hint:

The vector \( a\hat{i} + b\hat{j} + c\hat{k} \) directly gives the direction ratios \( \langle a, b, c \rangle \) for a line parallel to it. These values are the denominators in the Cartesian equation. Be careful with the signs of the coordinates in the numerators.

Answer 1

Step 1: Understanding the Concept:
The Cartesian equation of a line in 3D space is determined by a point on the line and its direction ratios. The direction ratios of the line will be the same as the components of the vector to which it is parallel.
Step 2: Key Formula or Approach:
The standard Cartesian (or symmetric) form of the equation of a line passing through the point \( (x_0, y_0, z_0) \) with direction ratios \( \langle a, b, c \rangle \) is: \[ \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} \] Step 3: Detailed Explanation or Calculation:
We are given: - The point on the line: \( (x_0, y_0, z_0) = (3, -2, -5) \). - The line is parallel to the vector \( 3\hat{i} + 2\hat{j} - 2\hat{k} \). This means the direction ratios of the line are the components of this vector: \( \langle a, b, c \rangle = \langle 3, 2, -2 \rangle \).
Now, substitute these values into the standard Cartesian form: \[ \frac{x - 3}{3} = \frac{y - (-2)}{2} = \frac{z - (-5)}{-2} \] \[ \frac{x - 3}{3} = \frac{y + 2}{2} = \frac{z + 5}{-2} \] Step 4: Final Answer:
The Cartesian equation of the line is \( \frac{x - 3}{3} = \frac{y + 2}{2} = \frac{z + 5}{-2} \).