Find the equation of line through `(1, 2, -1)` and perpendicular to each of the lines `(x)/(1)=(y)/(0)=(z)/(-1) and (x)/(3)=(y)/(4)=(z)/(5)`.

Answer

To find the equation of the line through the point \( (1, 2, -1) \) and perpendicular to the given lines, we will follow these steps:

### Step 1: Identify the direction ratios of the given lines
The first line is given by:
\[ \frac{x}{1} = \frac{y}{0} = \frac{z}{-1} \]
From this, we can extract the direction ratios as \( (1, 0, -1) \).

The second line is given by:
\[ \frac{x}{3} = \frac{y}{4} = \frac{z}{5} \]
From this, we can extract the direction ratios as \( (3, 4, 5) \).

### Step 2: Set up the equations for perpendicularity
Let the direction ratios of the required line be \( (a, b, c) \). Since the required line is perpendicular to both given lines, we can set up the following equations using the dot product:

1. For the first line:
\[ 1 \cdot a + 0 \cdot b - 1 \cdot c = 0 \implies a - c = 0 \implies a = c \]

2. For the second line:
\[ 3 \cdot a + 4 \cdot b + 5 \cdot c = 0 \]

### Step 3: Substitute \( c \) with \( a \)
From the first equation, we have \( c = a \). Substitute \( c \) in the second equation:
\[ 3a + 4b + 5a = 0 \implies 8a + 4b = 0 \implies 4b = -8a \implies b = -2a \]

### Step 4: Express direction ratios in terms of \( a \)
Now we have:
\[ a = a, \quad b = -2a, \quad c = a \]
Thus, the direction ratios can be expressed as:
\[ (a, -2a, a) = a(1, -2, 1) \]

### Step 5: Write the equation of the line
Using the point \( (1, 2, -1) \) and the direction ratios \( (1, -2, 1) \), the equation of the line can be written in symmetric form:
\[ \frac{x - 1}{1} = \frac{y - 2}{-2} = \frac{z + 1}{1} \]

### Final Equation
Thus, the equation of the required line is:
\[ \frac{x - 1}{1} = \frac{y - 2}{-2} = \frac{z + 1}{1} \]

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