Question

Find the vector equation of a straight line passing through the point (5, 2, -4) and parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).

Hint:

This is a direct application of a fundamental formula. Ensure you can distinguish between the position vector of the point (\(\vec{a}\)) and the direction vector (\(\vec{b}\)). The direction vector is always multiplied by the scalar parameter.

Answer 1

Step 1: Understanding the Concept:
The vector equation of a line describes the position vector \( \vec{r} \) of any point on the line. It is determined by a known point on the line and a direction vector that is parallel to the line.
Step 2: Key Formula or Approach:
The vector equation of a line passing through a point with position vector \( \vec{a} \) and parallel to a vector \( \vec{b} \) is given by: \[ \vec{r} = \vec{a} + \lambda \vec{b} \] where \( \lambda \) is a scalar parameter.
Step 3: Detailed Explanation:
From the problem statement, we identify the position vector of the given point and the parallel vector.
The line passes through the point (5, 2, -4). The position vector \( \vec{a} \) for this point is: \[ \vec{a} = 5\hat{i} + 2\hat{j} - 4\hat{k} \] The line is parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \). This is our direction vector \( \vec{b} \): \[ \vec{b} = 3\hat{i} + 2\hat{j} - 8\hat{k} \] Now, we substitute \( \vec{a} \) and \( \vec{b} \) into the standard formula: \[ \vec{r} = (5\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 8\hat{k}) \] This equation represents all points on the line for different values of \( \lambda \).
Step 4: Final Answer:
The required vector equation of the straight line is \( \vec{r} = (5\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 8\hat{k}) \).