Question

Let a = (−1, 3, 1) and b = (2, 4, 3). Then find projb
a
, proja
b
and compb
a

Answer 1

To find the projection of vector \(\mathbf{a}\) onto vector \(\mathbf{b}\) (\(\text{proj}_{\mathbf{b}} \mathbf{a}\)), the projection of vector \(\mathbf{b}\) onto vector \(\mathbf{a}\) (\(\text{proj}_{\mathbf{a}} \mathbf{b}\)), and the component of vector \(\mathbf{a}\) in the direction of vector \(\mathbf{b}\) (\(\text{comp}_{\mathbf{b}} \mathbf{a}\)), we can use the following formulas:

1. **Projection of \(\mathbf{a}\) onto \(\mathbf{b}\)**:

\[
\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}
\]

2. **Projection of \(\mathbf{b}\) onto \(\mathbf{a}\)**:

\[
\text{proj}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{b} \cdot \mathbf{a}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a}
\]

3. **Component of \(\mathbf{a}\) in the direction of \(\mathbf{b}\)**:

\[
\text{comp}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}
\]

Now let's compute these for \(\mathbf{a} = (-1, 3, 1)\) and \(\mathbf{b} = (2, 4, 3)\).

### Step 1: Compute \(\mathbf{a} \cdot \mathbf{b}\)

\[
\mathbf{a} \cdot \mathbf{b} = (-1)(2) + (3)(4) + (1)(3) = -2 + 12 + 3 = 13
\]

### Step 2: Compute \(\mathbf{b} \cdot \mathbf{b}\)

\[
\mathbf{b} \cdot \mathbf{b} = (2)(2) + (4)(4) + (3)(3) = 4 + 16 + 9 = 29
\]

### Step 3: Compute \(\mathbf{a} \cdot \mathbf{a}\)

\[
\mathbf{a} \cdot \mathbf{a} = (-1)(-1) + (3)(3) + (1)(1) = 1 + 9 + 1 = 11
\]

### Step 4: Compute \(\text{proj}_{\mathbf{b}} \mathbf{a}\)

\[
\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{13}{29} \mathbf{b} = \frac{13}{29} (2, 4, 3) = \left( \frac{26}{29}, \frac{52}{29}, \frac{39}{29} \right)
\]

### Step 5: Compute \(\text{proj}_{\mathbf{a}} \mathbf{b}\)

\[
\text{proj}_{\mathbf{a}} \mathbf{b} = \frac{13}{11} \mathbf{a} = \frac{13}{11} (-1, 3, 1) = \left( -\frac{13}{11}, \frac{39}{11}, \frac{13}{11} \right)
\]

### Step 6: Compute \(|\mathbf{b}|\)

\[
|\mathbf{b}| = \sqrt{\mathbf{b} \cdot \mathbf{b}} = \sqrt{29}
\]

### Step 7: Compute \(\text{comp}_{\mathbf{b}} \mathbf{a}\)

\[
\text{comp}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} = \frac{13}{\sqrt{29}}
\]

### Summary of Results

- \(\text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{26}{29}, \frac{52}{29}, \frac{39}{29} \right)\)
- \(\text{proj}_{\mathbf{a}} \mathbf{b} = \left( -\frac{13}{11}, \frac{39}{11}, \frac{13}{11} \right)\)
- \(\text{comp}_{\mathbf{b}} \mathbf{a} = \frac{13}{\sqrt{29}}\)