If the line `(x)/(1)=(y)/(2)=(z)/(3)` then convert this in a vector form

Answer

To convert the given line equation \(\frac{x}{1} = \frac{y}{2} = \frac{z}{3}\) into vector form, we can follow these steps:

### Step 1: Understand the given equation
The equation \(\frac{x}{1} = \frac{y}{2} = \frac{z}{3}\) represents a line in three-dimensional space. Here, the coefficients of \(x\), \(y\), and \(z\) indicate the direction ratios of the line.

### Step 2: Identify the direction ratios
From the equation, we can identify the direction ratios:
- For \(x\): 1
- For \(y\): 2
- For \(z\): 3

Thus, the direction ratios of the line are \( (1, 2, 3) \).

### Step 3: Determine a point on the line
The line passes through the origin, which can be represented as the point \( (0, 0, 0) \).

### Step 4: Write the vector equation of the line
The vector equation of a line can be expressed in the form:
\[ \vec{R} = \vec{A} + \lambda \vec{B} \]
where:
- \(\vec{R}\) is the position vector of any point on the line,
- \(\vec{A}\) is the position vector of a specific point on the line (in this case, the origin),
- \(\lambda\) is a scalar parameter,
- \(\vec{B}\) is the direction vector of the line.

### Step 5: Substitute the values into the equation
1. The position vector \(\vec{A}\) for the point \( (0, 0, 0) \) is:
\[ \vec{A} = 0 \hat{i} + 0 \hat{j} + 0 \hat{k} = \vec{0} \]
2. The direction vector \(\vec{B}\) based on the direction ratios is:
\[ \vec{B} = 1 \hat{i} + 2 \hat{j} + 3 \hat{k} \]

Now substituting these into the vector equation:
\[ \vec{R} = \vec{0} + \lambda (1 \hat{i} + 2 \hat{j} + 3 \hat{k}) \]
This simplifies to:
\[ \vec{R} = \lambda (1 \hat{i} + 2 \hat{j} + 3 \hat{k}) \]

### Final Answer
Thus, the vector form of the line is:
\[ \vec{R} = \lambda (1 \hat{i} + 2 \hat{j} + 3 \hat{k}) \]

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