Consider the vectors `a = i- 2j + k and b= 4i- 4j + 7k`
What is the scalar projection of a on b?

Options

A. 1
B. `(19)/(9)`
C. `(17)/(9)`
D. `(23)/(9)`

Answer

To find the scalar projection of vector **a** on vector **b**, we can use the formula:

\[ \text{Scalar Projection of } \mathbf{a} \text{ on } \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \]

Where:
- \(\mathbf{a} \cdot \mathbf{b}\) is the dot product of vectors **a** and **b**.
- \(|\mathbf{b}|\) is the magnitude (or length) of vector **b**.

### Step 1: Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\)

Given:
\[ \mathbf{a} = \mathbf{i} - 2\mathbf{j} + \mathbf{k} \]
\[ \mathbf{b} = 4\mathbf{i} - 4\mathbf{j} + 7\mathbf{k} \]

The dot product is calculated as follows:

\[ \mathbf{a} \cdot \mathbf{b} = (1)(4) + (-2)(-4) + (1)(7) \]
\[ = 4 + 8 + 7 \]
\[ = 19 \]

### Step 2: Calculate the magnitude of vector **b**

The magnitude of vector **b** is given by the formula:

\[ |\mathbf{b}| = \sqrt{(4^2) + (-4^2) + (7^2)} \]
\[ = \sqrt{16 + 16 + 49} \]
\[ = \sqrt{81} \]
\[ = 9 \]

### Step 3: Calculate the scalar projection

Now we can substitute the values into the scalar projection formula:

\[ \text{Scalar Projection of } \mathbf{a} \text{ on } \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} = \frac{19}{9} \]

### Final Answer

Thus, the scalar projection of vector **a** on vector **b** is:

\[ \frac{19}{9} \]

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