Question

Which of the following cannot be the direction ratios of the straight line \(\frac{x - 3}{2} = \frac{2 - y}{3} = \frac{z + 4}{-1}\)?

• A. \(2, 3, -1\)
• B. \(-2, 3, 1\)
• C. \(2, -3, -1\)
• D. \(6, -9, -3\)

Hint:

When finding the direction ratios of a line from its parametric form, look for the coefficients of \(x\), \(y\), and \(z\). These coefficients directly give you the direction ratios. Additionally, any scalar multiple of these direction ratios represents a line with the same direction, so negatives and multiples are valid answers. Always verify the sign and magnitude of direction ratios before concluding.

Answer 1

The Correct Option is C

The symmetric equation of the line is given by: \( \frac{x - 3}{2} = \frac{2 - y}{3} = \frac{z + 4}{-1} = k \), where \(k\) is a parameter.

This equation defines a line in 3D space. The direction ratios are derived from the denominators of the fractions, corresponding to the coefficients relative to the parameter \(k\). These direction ratios are \(2\), \(-3\), and \(-1\).

Direction ratios of a line in three dimensions are subject to arbitrary scalar multiplication. Consequently, any set of direction ratios that is a scalar multiple of \((2, -3, -1)\) is valid.

Let's evaluate the provided options:

1. \(2, -3, -1\): This matches the original direction ratios.
2. \(-2, 3, 1\): This is equivalent to the original set multiplied by \(-1\).
3. \(2, 3, -1\): This set cannot be a scalar multiple of the original. Altering the sign of only one component disrupts the proportional relationship required for direction ratios.
4. \(6, -9, -3\): This is the original set multiplied by \(3\).

Therefore, the direction ratios \(2, -3, -1\) cannot represent the line because they do not reflect the proportional transformations permissible for direction ratios derived from the symmetric equation.