Question

Which one of the following is a point on the straight line $\vec{r}=(13\hat{i}-14\hat{j}+23\hat{k})+\lambda(5\hat{i}-7\hat{j}-9\hat{k})$?

• A. (13, -14, -23)
• B. (5, -7, -9)
• C. (23, -28, 7)
• D. (23, -28, 5)
• E. (13, 14, 23)

Hint:

The starting point $(\lambda=0)$ is usually given in the first part of the vector equation.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The vector equation of a line generates the position vectors of all points on that line by varying the parameter \( \lambda \). To check if a given point lies on the line, we need to see if there exists a single value of \( \lambda \) that produces the coordinates of that point.
Step 2: Key Formula or Approach:
Any point \( (x, y, z) \) on the line can be represented as:
\[ x\hat{i} + y\hat{j} + z\hat{k} = (13\hat{i} - 14\hat{j} + 23\hat{k}) + \lambda(5\hat{i} - 7\hat{j} - 9\hat{k}) \] This gives us three parametric equations by equating the components:
\[ x = 13 + 5\lambda \] \[ y = -14 - 7\lambda \] \[ z = 23 - 9\lambda \] We will test each option by trying to solve for a consistent value of \( \lambda \).
Step 3: Detailed Explanation:
Let's test option (D): The point (23, -28, 5).
Substitute these coordinates into the parametric equations.
For x = 23:
\[ 23 = 13 + 5\lambda \] \[ 10 = 5\lambda \implies \lambda = 2 \] For y = -28:
\[ -28 = -14 - 7\lambda \] \[ -14 = -7\lambda \implies \lambda = 2 \] For z = 5:
\[ 5 = 23 - 9\lambda \] \[ -18 = -9\lambda \implies \lambda = 2 \] Since we found the same value of \( \lambda = 2 \) for all three coordinates, the point (23, -28, 5) lies on the given line.
Let's quickly check another option to see why it fails, for example, option (C): (23, -28, 7).
From the x and y coordinates, we already know we need \( \lambda=2 \). Let's check if this works for z=7.
\[ z = 23 - 9\lambda = 23 - 9(2) = 23 - 18 = 5 \] This gives z=5, but the point in option (C) has z=7. Since \( 5 \neq 7 \), the point (23, -28, 7) does not lie on the line.
Step 4: Final Answer:
The point (23, -28, 5) is on the straight line.