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The question asks to **prove that the orthogonal projection of vector \( \vec{A} \) onto vector \( \vec{B} \) is perpendicular to the projection of the component of \( \vec{A} \) that is orthogonal to \( \vec{B} \)**.

### Let's break this down:
1. **Orthogonal projection of \( \vec{A} \) onto \( \vec{B} \):**
   The projection of \( \vec{A} \) onto \( \vec{B} \), denoted as \( \text{Proj}_{\vec{B}} \vec{A} \), is given by:
   \[
   \text{Proj}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{\vec{B} \cdot \vec{B}} \vec{B}
   \]

2. **Component of \( \vec{A} \) orthogonal to \( \vec{B} \):**
   The component of \( \vec{A} \) that is orthogonal to \( \vec{B} \) is given by:
   \[
   \vec{A}_\perp = \vec{A} - \text{Proj}_{\vec{B}} \vec{A}
   \]
   This is the part of \( \vec{A} \) that does not lie along \( \vec{B} \), and it is perpendicular to \( \vec{B} \).

3. **Proving perpendicularity:**
   The dot product between the projection \( \text{Proj}_{\vec{B}} \vec{A} \) and the orthogonal component \( \vec{A}_\perp \) should be zero to prove they are perpendicular. We compute the dot product:
   \[
   \text{Proj}_{\vec{B}} \vec{A} \cdot \vec{A}_\perp = \left( \frac{\vec{A} \cdot \vec{B}}{\vec{B} \cdot \vec{B}} \vec{B} \right) \cdot \left( \vec{A} - \frac{\vec{A} \cdot \vec{B}}{\vec{B} \cdot \vec{B}} \vec{B} \right)
   \]
   Since \( \text{Proj}_{\vec{B}} \vec{A} \) is parallel to \( \vec{B} \) and \( \vec{A}_\perp \) is perpendicular to \( \vec{B} \), their dot product will be zero, confirming that they are perpendicular.

Thus, the projection of \( \vec{A} \) onto \( \vec{B} \) is perpendicular to the orthogonal component of \( \vec{A} \) with respect to \( \vec{B} \).

Would you like a deeper explanation or more details? Here are 5 related questions:

1. What is the geometric significance of vector projections in linear algebra?
2. How do we derive the formula for the orthogonal projection of one vector onto another?
3. What is the relationship between dot product and perpendicularity of vectors?
4. How can we use projections to decompose a vector into parallel and perpendicular components?
5. In what scenarios are vector projections commonly used in physics and engineering?

**Tip:** Remember that orthogonal projections are a fundamental tool in many optimization problems, including minimizing distances in vector spaces.

Buktikan bahwa proyeksi orthogonal dari A pada B saling tegak lurus dengan proyeksi dari A yang tegak lurus pada B!

This article provides a proof that the orthogonal projection of vector A onto vector B is perpendicular to the projection of the component of A that is orthogonal to B. Includes step-by-step calculations using vector projection formulas and the dot product theorem. Suitable for undergraduate students studying linear algebra.

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