Let `vecu, vecv and vecw` be such that `|vecu|=1,|vecv|=2 and |vecw|=3` if the projection of `vecv " along " vecu` is equal to that of `vecw` along `vecu` and vectors `vecv and vecw` are perpendicular to each other then `|vecu-vecv + vecw|` equals

Explanation

To solve the problem step by step, we will use the properties of vector projections and the given conditions.

### Step 1: Understand the Given Information
We have three vectors:
- \( \vec{u} \) such that \( |\vec{u}| = 1 \)
- \( \vec{v} \) such that \( |\vec{v}| = 2 \)
- \( \vec{w} \) such that \( |\vec{w}| = 3 \)

We know that the projection of \( \vec{v} \) along \( \vec{u} \) is equal to the projection of \( \vec{w} \) along \( \vec{u} \), and that \( \vec{v} \) and \( \vec{w} \) are perpendicular to each other.

### Step 2: Write the Projection Formulas
The projection of \( \vec{v} \) along \( \vec{u} \) is given by:
\[ \text{proj}_{\vec{u}} \vec{v} = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}|} \vec{u} \]
Since \( |\vec{u}| = 1 \), this simplifies to:
\[ \text{proj}_{\vec{u}} \vec{v} = \vec{u} \cdot \vec{v} \cdot \vec{u} \]

Similarly, the projection of \( \vec{w} \) along \( \vec{u} \) is:
\[ \text{proj}_{\vec{u}} \vec{w} = \vec{u} \cdot \vec{w} \cdot \vec{u} \]

### Step 3: Set the Projections Equal
Given that the projections are equal:
\[ \vec{u} \cdot \vec{v} = \vec{u} \cdot \vec{w} \]

### Step 4: Use the Perpendicular Condition
Since \( \vec{v} \) and \( \vec{w} \) are perpendicular, we have:
\[ \vec{v} \cdot \vec{w} = 0 \]

### Step 5: Calculate \( |\vec{u} - \vec{v} + \vec{w}| \)
We need to find \( |\vec{u} - \vec{v} + \vec{w}| \). We can use the formula for the magnitude of a vector:
\[ |\vec{u} - \vec{v} + \vec{w}|^2 = |\vec{u}|^2 + |\vec{v}|^2 + |\vec{w}|^2 - 2(\vec{u} \cdot \vec{v}) + 2(\vec{u} \cdot \vec{w}) - 2(\vec{v} \cdot \vec{w}) \]

### Step 6: Substitute the Magnitudes
Substituting the magnitudes:
- \( |\vec{u}|^2 = 1^2 = 1 \)
- \( |\vec{v}|^2 = 2^2 = 4 \)
- \( |\vec{w}|^2 = 3^2 = 9 \)

Now substituting into the equation:
\[ |\vec{u} - \vec{v} + \vec{w}|^2 = 1 + 4 + 9 - 2(\vec{u} \cdot \vec{v}) + 2(\vec{u} \cdot \vec{w}) - 0 \]

### Step 7: Simplify the Expression
Since \( \vec{u} \cdot \vec{v} = \vec{u} \cdot \vec{w} \), let \( k = \vec{u} \cdot \vec{v} = \vec{u} \cdot \vec{w} \):
\[ |\vec{u} - \vec{v} + \vec{w}|^2 = 14 - 2k + 2k = 14 \]

### Step 8: Final Calculation
Thus:
\[ |\vec{u} - \vec{v} + \vec{w}| = \sqrt{14} \]

### Final Answer
\[ |\vec{u} - \vec{v} + \vec{w}| = \sqrt{14} \]