Write the vector equation of the line :
`(x - 5)/(3) = (y + 4)/(7) = (6 -z)/(2)` .

Answer

To write the vector equation of the line given by the equation

\[ \frac{x - 5}{3} = \frac{y + 4}{7} = \frac{6 - z}{2}, \]

we will follow these steps:

### Step 1: Rewrite the equation
First, we can rewrite the equation in a more standard form. Notice that we can express \(6 - z\) as \(- (z - 6)\). Thus, we can rewrite the equation as:

\[ \frac{x - 5}{3} = \frac{y + 4}{7} = \frac{-(z - 6)}{2}. \]

This helps to keep the variables in a positive format.

### Step 2: Identify the point on the line
From the equation, we can determine a point through which the line passes. We can set the common ratio equal to a parameter \(t\):

1. For \(x\):
\[ x - 5 = 3t \implies x = 3t + 5. \]

2. For \(y\):
\[ y + 4 = 7t \implies y = 7t - 4. \]

3. For \(z\):
\[ -(z - 6) = 2t \implies z = 6 - 2t. \]

Now, we can find the coordinates of a specific point on the line by setting \(t = 0\):
- When \(t = 0\):
- \(x = 5\)
- \(y = -4\)
- \(z = 6\)

Thus, the point \(P\) is \((5, -4, 6)\).

### Step 3: Identify the direction ratios
The direction ratios can be directly taken from the denominators of the fractions in the original equation. The direction ratios are:
- For \(x\): \(3\)
- For \(y\): \(7\)
- For \(z\): \(-2\) (since we have \(6 - z\), we take it as negative)

Thus, the direction ratios are \((3, 7, -2)\).

### Step 4: Write the vector equation
The vector equation of the line can be expressed in the form:

\[ \mathbf{r} = \mathbf{p} + \lambda \mathbf{d}, \]

where \(\mathbf{p}\) is the position vector of the point \(P\) and \(\mathbf{d}\) is the direction vector.

The position vector \(\mathbf{p}\) corresponding to the point \(P(5, -4, 6)\) is:

\[ \mathbf{p} = 5\mathbf{i} - 4\mathbf{j} + 6\mathbf{k}. \]

The direction vector \(\mathbf{d}\) corresponding to the direction ratios \((3, 7, -2)\) is:

\[ \mathbf{d} = 3\mathbf{i} + 7\mathbf{j} - 2\mathbf{k}. \]

Putting it all together, the vector equation of the line is:

\[ \mathbf{r} = (5\mathbf{i} - 4\mathbf{j} + 6\mathbf{k}) + \lambda (3\mathbf{i} + 7\mathbf{j} - 2\mathbf{k}). \]

### Final Answer
Thus, the vector equation of the line is:

\[ \mathbf{r} = (5 + 3\lambda)\mathbf{i} + (-4 + 7\lambda)\mathbf{j} + (6 - 2\lambda)\mathbf{k}. \]

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