Which of the following triplets give the direction cosines of a line ?

Options

A. 1,1,1
B. 1,-1,1
C. 1,1,-1
D. `1/(sqrt""3),1/(sqrt""3),1/(sqrt""3)`

Answer

To determine which triplet gives the direction cosines of a line, we need to check each option against the condition that the sum of the squares of the direction cosines equals 1. The direction cosines of a line are usually represented as \( L, M, N \), and the condition can be expressed mathematically as:

\[ L^2 + M^2 + N^2 = 1 \]

Now, let's evaluate each option step by step:

### Step 1: Evaluate Option 1: (1, 1, 1)
- Calculate the sum of squares:
\[ L^2 + M^2 + N^2 = 1^2 + 1^2 + 1^2 = 1 + 1 + 1 = 3 \]
- Since \( 3 \neq 1 \), this option does not satisfy the condition.

### Step 2: Evaluate Option 2: (1, -1, 1)
- Calculate the sum of squares:
\[ L^2 + M^2 + N^2 = 1^2 + (-1)^2 + 1^2 = 1 + 1 + 1 = 3 \]
- Since \( 3 \neq 1 \), this option does not satisfy the condition.

### Step 3: Evaluate Option 3: (1, 1, -1)
- Calculate the sum of squares:
\[ L^2 + M^2 + N^2 = 1^2 + 1^2 + (-1)^2 = 1 + 1 + 1 = 3 \]
- Since \( 3 \neq 1 \), this option does not satisfy the condition.

### Step 4: Evaluate Option 4: \(\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)\)
- Calculate the sum of squares:
\[ L^2 + M^2 + N^2 = \left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{3}{3} = 1 \]
- Since \( 1 = 1 \), this option satisfies the condition.

### Conclusion:
The only triplet that gives the direction cosines of a line is option 4: \(\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)\).

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