What is the projection of the vector `hat(i)-2hat(j)-hat(k)` on the vector `4hat(i)-4hat(j)+7hat(k)`?

Options

A. `(sqrt(5))/(2)`
B. `(19)/(9)`
C. `(sqrt(5))/(4)`
D. `(11)/(3)`

Explanation

To find the projection of the vector \( \hat{i} - 2\hat{j} - \hat{k} \) on the vector \( 4\hat{i} - 4\hat{j} + 7\hat{k} \), we can follow these steps:

### Step 1: Define the vectors
Let:
- \( \mathbf{a} = \hat{i} - 2\hat{j} - \hat{k} \)
- \( \mathbf{b} = 4\hat{i} - 4\hat{j} + 7\hat{k} \)

### Step 2: Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \)
The dot product is calculated as follows:
\[ \mathbf{a} \cdot \mathbf{b} = (1)(4) + (-2)(-4) + (-1)(7) \]
Calculating each term:
- \( 1 \cdot 4 = 4 \)
- \( -2 \cdot -4 = 8 \)
- \( -1 \cdot 7 = -7 \)

Now, sum these results:
\[ \mathbf{a} \cdot \mathbf{b} = 4 + 8 - 7 = 5 \]

### Step 3: Calculate the magnitude of vector \( \mathbf{b} \)
The magnitude of \( \mathbf{b} \) is given by:
\[ |\mathbf{b}| = \sqrt{(4)^2 + (-4)^2 + (7)^2} \]
Calculating each square:
- \( 4^2 = 16 \)
- \( (-4)^2 = 16 \)
- \( 7^2 = 49 \)

Now sum these:
\[ |\mathbf{b}| = \sqrt{16 + 16 + 49} = \sqrt{81} = 9 \]

### Step 4: Calculate the projection of \( \mathbf{a} \) on \( \mathbf{b} \)
The projection of \( \mathbf{a} \) on \( \mathbf{b} \) is given by the formula:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \mathbf{b} \]
First, we need \( |\mathbf{b}|^2 \):
\[ |\mathbf{b}|^2 = 9^2 = 81 \]

Now substitute the values:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{5}{81} \mathbf{b} \]
Substituting \( \mathbf{b} \):
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{5}{81} (4\hat{i} - 4\hat{j} + 7\hat{k}) = \left(\frac{20}{81}\hat{i} - \frac{20}{81}\hat{j} + \frac{35}{81}\hat{k}\right) \]

### Final Answer
The projection of the vector \( \hat{i} - 2\hat{j} - \hat{k} \) on the vector \( 4\hat{i} - 4\hat{j} + 7\hat{k} \) is:
\[ \frac{20}{81}\hat{i} - \frac{20}{81}\hat{j} + \frac{35}{81}\hat{k} \]