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

Options

A. `1//5`
B. `1//sqrt5`
C. `1//29`
D. `1 sqrt29`

Answer

To solve for the scalar \( m \) such that \( \vec{b} = m \vec{a} \) where \( \vec{b} \) is a unit vector co-directional with \( \vec{a} \), we can follow these steps:

### Step 1: Write down the given vector \( \vec{a} \)
We have:
\[ \vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \]

### Step 2: Find the magnitude of \( \vec{a} \)
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 \( \vec{a} \):
\[ |\vec{a}| = \sqrt{2^2 + (-3)^2 + 4^2} = \sqrt{4 + 9 + 16} = \sqrt{29} \]

### Step 3: Understand the properties of \( \vec{b} \)
Since \( \vec{b} \) is a unit vector co-directional with \( \vec{a} \), we have:
\[ |\vec{b}| = 1 \]
And from the relationship \( \vec{b} = m \vec{a} \), we can take the magnitude of both sides:
\[ |\vec{b}| = |m| |\vec{a}| \]

### Step 4: Set up the equation
Substituting the known values:
\[ 1 = |m| \cdot |\vec{a}| \]
This simplifies to:
\[ 1 = |m| \cdot \sqrt{29} \]

### Step 5: Solve for \( m \)
To isolate \( m \), we rearrange the equation:
\[ |m| = \frac{1}{\sqrt{29}} \]
Since \( m \) is a scalar, we can take \( m \) to be positive (as \( \vec{b} \) is co-directional with \( \vec{a} \)):
\[ m = \frac{1}{\sqrt{29}} \]

### Final Answer
Thus, the value of \( m \) is:
\[ m = \frac{1}{\sqrt{29}} \]

---