If the direction ratios of two lines are (1,2,4) and (-1,-2,-3) then the acute angle between them is

Options

A. `cos^(-1)((-17)/(7sqrt(6)))`
B. `cos^(-1)((17)/(7sqrt(6)))`
C. `cos^(-1)((17)/(7))`
D. None of these

Explanation

To find the acute angle between two lines with given direction ratios, we can follow these steps:

### Step 1: Identify the Direction Ratios
The direction ratios of the first line are given as \( (1, 2, 4) \) and for the second line as \( (-1, -2, -3) \).

### Step 2: Use the Formula for Cosine of the Angle Between Two Lines
The cosine of the angle \( \theta \) between two lines can be calculated using the formula:
\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \]
where \( (a_1, b_1, c_1) \) are the direction ratios of the first line and \( (a_2, b_2, c_2) \) are the direction ratios of the second line.

### Step 3: Calculate the Dot Product
Calculate the dot product \( a_1 a_2 + b_1 b_2 + c_1 c_2 \):
\[ 1 \cdot (-1) + 2 \cdot (-2) + 4 \cdot (-3) = -1 - 4 - 12 = -17 \]

### Step 4: Calculate the Magnitudes of the Direction Ratios
Calculate the magnitudes of the direction ratios:
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{1^2 + 2^2 + 4^2} = \sqrt{1 + 4 + 16} = \sqrt{21} \]
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{(-1)^2 + (-2)^2 + (-3)^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

### Step 5: Substitute Values into the Cosine Formula
Substituting the values we calculated into the cosine formula:
\[ \cos \theta = \frac{-17}{\sqrt{21} \cdot \sqrt{14}} = \frac{-17}{\sqrt{294}} \]
Since we need the acute angle, we take the absolute value:
\[ \cos \theta = \frac{17}{\sqrt{294}} \]

### Step 6: Calculate the Angle
To find \( \theta \), take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{17}{\sqrt{294}}\right) \]

### Final Result
Thus, the acute angle \( \theta \) between the two lines is given by:
\[ \theta = \cos^{-1}\left(\frac{17}{\sqrt{294}}\right) \]