Given `vec(a)=2hat(i)-3hat(j)+4hat(k) and hat(b)` is a unit vector codirectional with `hat(a)`. If m is a scalar such that `hat(b)=m vec(a)`, then what is the value of m ?

Options

A. `1//5`
B. `1//sqrt(5)`
C. `1//29`
D. `1//sqrt(29)`

Explanation

To solve the problem, we need to find the scalar \( m \) such that the unit vector \( \hat{b} \) is equal to \( m \vec{a} \), where \( \vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k} \).

### Step-by-Step Solution:

1. **Identify the Given Vector**:
\[ \vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k} \]

2. **Find the Magnitude of Vector \( \vec{a} \)**:
The magnitude of a vector \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) is given by:
\[ |\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]
For our vector \( \vec{a} \):
\[ |\vec{a}| = \sqrt{(2)^2 + (-3)^2 + (4)^2} = \sqrt{4 + 9 + 16} = \sqrt{29} \]

3. **Understanding the Unit Vector \( \hat{b} \)**:
Since \( \hat{b} \) is a unit vector co-directional with \( \vec{a} \), its magnitude is 1:
\[ |\hat{b}| = 1 \]

4. **Relate \( \hat{b} \) and \( \vec{a} \)**:
We know that \( \hat{b} = m \vec{a} \). Taking the magnitude of both sides:
\[ |\hat{b}| = |m| |\vec{a}| \]
Substituting the known magnitudes:
\[ 1 = |m| \cdot \sqrt{29} \]

5. **Solve for \( m \)**:
Rearranging the equation gives:
\[ |m| = \frac{1}{\sqrt{29}} \]
Since \( m \) can be positive or negative, we take the positive value for the unit vector:
\[ m = \frac{1}{\sqrt{29}} \]

6. **Conclusion**:
The value of \( m \) is:
\[ m = \frac{1}{\sqrt{29}} \]