The vector component of vector `vec(A) = 3 hat(i) + 4 hat(j) + 5 hat(k)` along vector `vec(B) = hat(i) + hat(j) + hat(k)` is

Options

A. `2 hat (i) + 2 hat(j) + 2 hat(k)`
B. `3 hat (i) + 3 hat(j) + 3 hat(k)`
C. `4 hat (i) + 4 hat(j) + 4 hat(k)`
D. `5 hat (i) + 5 hat(j) + 5 hat(k)`

Explanation

To find the vector component of vector \(\vec{A}\) along vector \(\vec{B}\), we can use the formula for the projection of one vector onto another. The formula for the projection of vector \(\vec{A}\) onto vector \(\vec{B}\) is given by:

\[ \text{proj}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{\vec{B} \cdot \vec{B}} \vec{B} \]

Where:
- \(\vec{A} \cdot \vec{B}\) is the dot product of vectors \(\vec{A}\) and \(\vec{B}\).
- \(\vec{B} \cdot \vec{B}\) is the dot product of vector \(\vec{B}\) with itself.

### Step 1: Calculate the dot product \(\vec{A} \cdot \vec{B}\)

Given:
\[ \vec{A} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \]
\[ \vec{B} = \hat{i} + \hat{j} + \hat{k} \]

The dot product \(\vec{A} \cdot \vec{B}\) is calculated as follows:

\[ \vec{A} \cdot \vec{B} = (3)(1) + (4)(1) + (5)(1) = 3 + 4 + 5 = 12 \]

### Step 2: Calculate the dot product \(\vec{B} \cdot \vec{B}\)

Now, we calculate \(\vec{B} \cdot \vec{B}\):

\[ \vec{B} \cdot \vec{B} = (1)(1) + (1)(1) + (1)(1) = 1 + 1 + 1 = 3 \]

### Step 3: Substitute into the projection formula

Now we can substitute the values into the projection formula:

\[ \text{proj}_{\vec{B}} \vec{A} = \frac{12}{3} \vec{B} = 4 \vec{B} \]

### Step 4: Calculate the vector component

Now substituting \(\vec{B}\):

\[ \text{proj}_{\vec{B}} \vec{A} = 4(\hat{i} + \hat{j} + \hat{k}) = 4\hat{i} + 4\hat{j} + 4\hat{k} \]

### Final Answer

The vector component of vector \(\vec{A}\) along vector \(\vec{B}\) is:

\[ \text{proj}_{\vec{B}} \vec{A} = 4\hat{i} + 4\hat{j} + 4\hat{k} \]

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