What is the angle between the lines whose direction cosines are proportional to `(2,3,4) and (1,-2,1)` respectively

Options

A. `90^@`
B. `60^@`
C. `45^@`
D. `30^@`

Answer

To find the angle between the lines whose direction cosines are proportional to the vectors \( (2, 3, 4) \) and \( (1, -2, 1) \), we can follow these steps:

### Step 1: Identify the Direction Cosines
Let the direction cosines of the first line be \( (a_1, b_1, c_1) = (2, 3, 4) \) and for the second line be \( (a_2, b_2, c_2) = (1, -2, 1) \).

### Step 2: Use the Dot Product Formula
The angle \( \theta \) between two lines can be found using the dot product formula:
\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \]

### Step 3: Calculate the Dot Product
Calculate \( a_1 a_2 + b_1 b_2 + c_1 c_2 \):
\[ = (2)(1) + (3)(-2) + (4)(1) = 2 - 6 + 4 = 0 \]

### Step 4: Analyze the Result
Since the dot product is zero, it implies that the two lines are perpendicular to each other. Therefore, the angle \( \theta \) between the lines is:
\[ \theta = 90^\circ \]

### Conclusion
The angle between the lines whose direction cosines are proportional to \( (2, 3, 4) \) and \( (1, -2, 1) \) is \( 90^\circ \).