How do you find the orthogonal projection of vector a onto vector b using vectors a = (3, 12) and b = (6, 9)?

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To find the orthogonal projection of vector a onto vector b, we follow a systematic approach using vector algebra.

The vectors given are:

  • a = (3, 12)
  • b = (6, 9)

1. **Calculate the dot product of a and b**: The dot product (denoted as a · b) is calculated as:

a · b = (3 * 6) + (12 * 9) = 18 + 108 = 126

2. **Calculate the dot product of b with itself**: To find the magnitude of vector b, we need b · b:

b · b = (6 * 6) + (9 * 9) = 36 + 81 = 117

3. **Use the projection formula**: The orthogonal projection of vector a onto vector b is given by the formula:

orth_ab = (a · b) / (b · b) * b

Now, substituting the values we calculated:

orth_ab = (126) / (117) * b

This simplifies to:

orth_ab = (126 / 117) * (6, 9)

4. **Calculating the scalar multiplication**:

orth_ab ≈ (1.0789) * (6, 9) = (6.4734, 9.7104)

5. **Final Result**: Thus, the orthogonal projection of vector a onto vector b is approximately:

orth_ab ≈ (6.47, 9.71)

This result provides a clear understanding of how vector a is projected onto vector b, offering insight into their spatial relationship in a 2D space.

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