What are the direction cosines of a line whose direction ratios are 3,4,12?

Options

A. `3/13, 4/13, 12/13`
B. `(-3)/13, (-4)/13, (-12)/13`
C. `1/13, 2/13, 3/13`
D. `(-1)/13, (-2)/13, (-3)/13`

Answer

To find the direction cosines of a line whose direction ratios are given as 3, 4, and 12, we can follow these steps:

### Step 1: Understand the relationship between direction ratios and direction cosines
Direction cosines of a line are defined as the cosines of the angles that the line makes with the coordinate axes. If the direction ratios of a line are given as \( a, b, c \), then the direction cosines \( l, m, n \) can be calculated using the formulas:
\[ l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}} \]

### Step 2: Identify the direction ratios
Given direction ratios are:
- \( a = 3 \)
- \( b = 4 \)
- \( c = 12 \)

### Step 3: Calculate \( a^2 + b^2 + c^2 \)
Now, we need to calculate \( a^2 + b^2 + c^2 \):
\[ a^2 = 3^2 = 9 \]
\[ b^2 = 4^2 = 16 \]
\[ c^2 = 12^2 = 144 \]
Adding these together:
\[ a^2 + b^2 + c^2 = 9 + 16 + 144 = 169 \]

### Step 4: Calculate the square root
Next, we find the square root of \( a^2 + b^2 + c^2 \):
\[ \sqrt{a^2 + b^2 + c^2} = \sqrt{169} = 13 \]

### Step 5: Calculate the direction cosines
Now we can find the direction cosines:
\[ l = \frac{a}{\sqrt{a^2 + b^2 + c^2}} = \frac{3}{13} \]
\[ m = \frac{b}{\sqrt{a^2 + b^2 + c^2}} = \frac{4}{13} \]
\[ n = \frac{c}{\sqrt{a^2 + b^2 + c^2}} = \frac{12}{13} \]

### Step 6: Consider the opposite direction
Direction cosines can also be in the opposite direction. Therefore, we can express the direction cosines as:
\[ l = \pm \frac{3}{13}, \quad m = \pm \frac{4}{13}, \quad n = \pm \frac{12}{13} \]

### Final Answer
Thus, the direction cosines of the line are:
\[ \left( \frac{3}{13}, \frac{4}{13}, \frac{12}{13} \right) \quad \text{and} \quad \left( -\frac{3}{13}, -\frac{4}{13}, -\frac{12}{13} \right) \]