Question

Assertion (A): A line can have direction cosines $\langle 1,1,1\rangle$.
Reason (R): $\cos\theta=1$ is possible for $\theta=0^\circ$.

• A. Both (A) and (R) are true and (R) is the correct explanation of (A).
• B. Both (A) and (R) are true but (R) is not the correct explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Hint:

Direction cosines must always satisfy the identity \(l^2+m^2+n^2=1\).

Answer 1

The Correct Option is D

The given question involves determining the truth of an assertion and a reason related to the concept of direction cosines in three-dimensional geometry.

Assertion (A): A line can have direction cosines \(\langle 1,1,1 \rangle\).

Reason (R): \(\cos\theta=1\) is possible for \(\theta=0^\circ\).

First, let's understand each part:

1. Direction Cosines: For a line in three-dimensional space, the direction cosines are the cosines of the angles that the line makes with the positive x, y, and z-axes. If a line has direction cosines \(l, m,\) and \(n\), they must satisfy the equation: \(l^2 + m^2 + n^2 = 1\). This is akin to the Pythagorean theorem applied to directions in space.
2. In the assertion, it is stated that direction cosines are \(\langle 1,1,1 \rangle\). Calculating the sum of squares: \(1^2 + 1^2 + 1^2 = 3\). This does not equal 1, therefore such direction cosines cannot describe any line.
3. Cosine of Angle: The given reason states that \(\cos\theta = 1\) is possible for \(\theta = 0^\circ\). This is true because the cosine of zero degrees is indeed 1.

To summarize, the assertion is false because there cannot be a line with direction cosines \(\langle 1,1,1 \rangle\) as they do not satisfy the required condition of the sum of squares equal to 1. However, the reason is true as it correctly describes a possible condition for the cosine function.

Conclusion: The correct answer is:

(A) is false but (R) is true.