Let ` vec u , vec va n d vec w` be such that `| vec u|=1,| vec v|=2a n d| vec w|=3.` If the projection of ` vec v` along ` vec u` is equal to that of ` vec w` along ` vec u` and vectors ` vec va n d vec w` are perpendicular to each other, then `| vec u- vec v+ vec w|` equals `2` b. `sqrt(7)` c. `sqrt(14)` d. `14`

Options

A. 2
B. sqrt(7)`
C. sqrt(14)`
D. 14`

Answer

To solve the problem step by step, we will analyze the given information and apply the necessary mathematical concepts.

### Given:
1. \( |\vec{u}| = 1 \)
2. \( |\vec{v}| = 2 \)
3. \( |\vec{w}| = 3 \)
4. The projection of \( \vec{v} \) along \( \vec{u} \) is equal to the projection of \( \vec{w} \) along \( \vec{u} \).
5. Vectors \( \vec{v} \) and \( \vec{w} \) are perpendicular to each other.

### Step 1: Write the projection formulas
The projection of vector \( \vec{a} \) along vector \( \vec{b} \) is given by:
\[ \text{Projection of } \vec{a} \text{ along } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \]
Thus, we can write:
\[ \text{Projection of } \vec{v} \text{ along } \vec{u} = \frac{\vec{v} \cdot \vec{u}}{|\vec{u}|} \]
\[ \text{Projection of } \vec{w} \text{ along } \vec{u} = \frac{\vec{w} \cdot \vec{u}}{|\vec{u}|} \]

### Step 2: Set the projections equal
Since the projections are equal:
\[ \frac{\vec{v} \cdot \vec{u}}{|\vec{u}|} = \frac{\vec{w} \cdot \vec{u}}{|\vec{u}|} \]
Given that \( |\vec{u}| = 1 \), this simplifies to:
\[ \vec{v} \cdot \vec{u} = \vec{w} \cdot \vec{u} \tag{1} \]

### Step 3: Use the perpendicularity condition
Since \( \vec{v} \) and \( \vec{w} \) are perpendicular:
\[ \vec{v} \cdot \vec{w} = 0 \tag{2} \]

### Step 4: Find \( |\vec{u} - \vec{v} + \vec{w}| \)
We want to find:
\[ |\vec{u} - \vec{v} + \vec{w}| \]
We will square this expression:
\[ |\vec{u} - \vec{v} + \vec{w}|^2 = (\vec{u} - \vec{v} + \vec{w}) \cdot (\vec{u} - \vec{v} + \vec{w}) \]
Expanding this:
\[ = \vec{u} \cdot \vec{u} - 2 \vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{v} + 2 \vec{u} \cdot \vec{w} - 2 \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{w} \]

### Step 5: Substitute known values
Substituting the magnitudes:
- \( \vec{u} \cdot \vec{u} = 1^2 = 1 \)
- \( \vec{v} \cdot \vec{v} = 2^2 = 4 \)
- \( \vec{w} \cdot \vec{w} = 3^2 = 9 \)

Using equations (1) and (2):
- From (1), \( \vec{u} \cdot \vec{v} = \vec{u} \cdot \vec{w} \) (let's denote this common value as \( k \))
- From (2), \( \vec{v} \cdot \vec{w} = 0 \)

Thus, we have:
\[ |\vec{u} - \vec{v} + \vec{w}|^2 = 1 - 2k + 4 + 2k + 9 \]
This simplifies to:
\[ = 14 + 0 = 14 \]

### Step 6: Find the magnitude
Taking the square root gives:
\[ |\vec{u} - \vec{v} + \vec{w}| = \sqrt{14} \]

### Final Answer
Thus, the final answer is:
\[ |\vec{u} - \vec{v} + \vec{w}| = \sqrt{14} \]