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

Question

a
, proja
b
and compb
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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

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