If a,b and c are three mutually perpendicular vectors, then the projection of the vectors
`l(a)/(|a|)+m(b)/(|b|)+n((axxb))/(|axxb|)` along the angle bisector of the vectors a and b is

Options

A. `(l+m)/(sqrt(2))`
B. `sqrt(l^(2)+m^(2)+n^(2))`
C. `(sqrt(l^(2)+m^(2)))/(sqrt(l^(2)+m^(2)+b^(2)))`
D. none of these

Answer

To solve the problem, we need to find the projection of the vector \( \frac{l \mathbf{a}}{|\mathbf{a}|} + \frac{m \mathbf{b}}{|\mathbf{b}|} + \frac{n (\mathbf{a} \times \mathbf{b})}{|\mathbf{a} \times \mathbf{b}|} \) along the angle bisector of the vectors \( \mathbf{a} \) and \( \mathbf{b} \).

### Step 1: Determine the Angle Bisector
The angle bisector \( \mathbf{y} \) of the vectors \( \mathbf{a} \) and \( \mathbf{b} \) can be expressed as:
\[ \mathbf{y} = \frac{\lambda \mathbf{a}}{|\mathbf{a}|} + \frac{\mathbf{b}}{|\mathbf{b}|} \]
where \( \lambda \) is a scalar that can be determined based on the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \).

### Step 2: Define the Vector to be Projected
Let \( \mathbf{x} = \frac{l \mathbf{a}}{|\mathbf{a}|} + \frac{m \mathbf{b}}{|\mathbf{b}|} + \frac{n (\mathbf{a} \times \mathbf{b})}{|\mathbf{a} \times \mathbf{b}|} \).

### Step 3: Calculate the Dot Product \( \mathbf{x} \cdot \mathbf{y} \)
To find the projection of \( \mathbf{x} \) onto \( \mathbf{y} \), we first need to calculate the dot product \( \mathbf{x} \cdot \mathbf{y} \):
\[ \mathbf{x} \cdot \mathbf{y} = \left( \frac{l \mathbf{a}}{|\mathbf{a}|} + \frac{m \mathbf{b}}{|\mathbf{b}|} + \frac{n (\mathbf{a} \times \mathbf{b})}{|\mathbf{a} \times \mathbf{b}|} \right) \cdot \left( \frac{\lambda \mathbf{a}}{|\mathbf{a}|} + \frac{\mathbf{b}}{|\mathbf{b}|} \right) \]
Expanding this gives:
\[ \mathbf{x} \cdot \mathbf{y} = \frac{l \lambda |\mathbf{a}|^2}{|\mathbf{a}|^2} + \frac{m}{|\mathbf{b}|} + \text{other terms} \]
Since \( \mathbf{a} \) and \( \mathbf{b} \) are mutually perpendicular, the cross product term will contribute zero.

### Step 4: Simplifying the Dot Product
The dot product simplifies to:
\[ \mathbf{x} \cdot \mathbf{y} = l \lambda + m \]

### Step 5: Calculate the Magnitude of \( \mathbf{y} \)
Next, we calculate the magnitude of \( \mathbf{y} \):
\[ |\mathbf{y}| = \sqrt{\left( \frac{\lambda}{|\mathbf{a}|} \right)^2 + \left( \frac{1}{|\mathbf{b}|} \right)^2} \]
This can be simplified to:
\[ |\mathbf{y}| = \lambda \sqrt{2} \]

### Step 6: Calculate the Projection
The scalar projection of \( \mathbf{x} \) onto \( \mathbf{y} \) is given by:
\[ \text{Projection} = \frac{\mathbf{x} \cdot \mathbf{y}}{|\mathbf{y}|} \]
Substituting the values we found:
\[ \text{Projection} = \frac{l \lambda + m}{\lambda \sqrt{2}} = \frac{l + m}{\sqrt{2}} \]

### Final Result
Thus, the projection of the vector \( \mathbf{x} \) along the angle bisector of vectors \( \mathbf{a} \) and \( \mathbf{b} \) is:
\[ \frac{l + m}{\sqrt{2}} \]