The angle between the straight lines
`(x+1)/(2)=(y-2)/(5)=(z+3)/(4)`
and `(x-1)/(1)=(y+2)/(2)=(z-3)/(-3)` is

Options

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

Answer

To find the angle between the two straight lines given by the equations:

1. \(\frac{x+1}{2} = \frac{y-2}{5} = \frac{z+3}{4}\)
2. \(\frac{x-1}{1} = \frac{y+2}{2} = \frac{z-3}{-3}\)

we can follow these steps:

### Step 1: Identify the direction ratios of the lines

The first line can be expressed in the form:
\[ \frac{x - (-1)}{2} = \frac{y - 2}{5} = \frac{z - (-3)}{4} \]
From this, we can identify the direction ratios \(d_1\) of the first line as:
\[ d_1 = (2, 5, 4) \]

The second line can be expressed as:
\[ \frac{x - 1}{1} = \frac{y - (-2)}{2} = \frac{z - 3}{-3} \]
From this, we can identify the direction ratios \(d_2\) of the second line as:
\[ d_2 = (1, 2, -3) \]

### Step 2: Calculate the dot product of the direction ratios

The dot product \(d_1 \cdot d_2\) is calculated as follows:
\[ d_1 \cdot d_2 = (2)(1) + (5)(2) + (4)(-3) \]
Calculating each term:
\[ = 2 + 10 - 12 = 0 \]

### Step 3: Calculate the magnitudes of the direction ratios

The magnitude of \(d_1\) is:
\[ |d_1| = \sqrt{2^2 + 5^2 + 4^2} = \sqrt{4 + 25 + 16} = \sqrt{45} \]

The magnitude of \(d_2\) is:
\[ |d_2| = \sqrt{1^2 + 2^2 + (-3)^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

### Step 4: Use the formula for the cosine of the angle between the lines

The cosine of the angle \(\theta\) between the two lines is given by:
\[ \cos \theta = \frac{d_1 \cdot d_2}{|d_1| |d_2|} \]
Substituting the values we calculated:
\[ \cos \theta = \frac{0}{\sqrt{45} \cdot \sqrt{14}} = 0 \]

### Step 5: Determine the angle \(\theta\)

Since \(\cos \theta = 0\), this means:
\[ \theta = \frac{\pi}{2} \text{ or } 90^\circ \]

### Final Answer
The angle between the two straight lines is \(90^\circ\).

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