If the components of `vec(b)` along and perpendicular to `vec(a)` are `lambdavec(a ) and vec(b)-lambdavec(a)` respectively, what is `lambda` equal to ?

Options

A. `(vec(a).vec(b))/(|vec(a)|)`
B. `(vec(a).vec(b))/(|vec(b)|)`
C. `(vec(a).vec(b))/(|vec(a)|^(2))`
D. `(vec(a).vec(b))/(|vec(b)|^(2))`

Explanation

To solve the problem, we need to find the value of \( \lambda \) given that the components of \( \vec{b} \) along and perpendicular to \( \vec{a} \) are \( \lambda \vec{a} \) and \( \vec{b} - \lambda \vec{a} \) respectively.

### Step-by-step Solution:

1. **Understanding the Components**:
- The component of \( \vec{b} \) along \( \vec{a} \) is given by the projection formula:
\[ \text{Projection of } \vec{b} \text{ on } \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \vec{a} \]
- According to the problem, this projection is equal to \( \lambda \vec{a} \).

2. **Setting Up the Equation**:
- From the projection formula, we can equate:
\[ \lambda \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \vec{a} \]

3. **Dividing Both Sides by \( \vec{a} \)**:
- Since \( \vec{a} \) is not the zero vector, we can divide both sides by \( \vec{a} \):
\[ \lambda = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \]

4. **Conclusion**:
- Thus, we find that:
\[ \lambda = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \]

### Final Answer:
\[ \lambda = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \]