Math Problem Statement

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

Solution

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

Let's break this down:

  1. Orthogonal projection of A\vec{A} onto B\vec{B}: The projection of A\vec{A} onto B\vec{B}, denoted as ProjBA\text{Proj}_{\vec{B}} \vec{A}, is given by: ProjBA=ABBBB\text{Proj}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{\vec{B} \cdot \vec{B}} \vec{B}

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

  3. Proving perpendicularity: The dot product between the projection ProjBA\text{Proj}_{\vec{B}} \vec{A} and the orthogonal component A\vec{A}_\perp should be zero to prove they are perpendicular. We compute the dot product: ProjBAA=(ABBBB)(AABBBB)\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 ProjBA\text{Proj}_{\vec{B}} \vec{A} is parallel to B\vec{B} and A\vec{A}_\perp is perpendicular to B\vec{B}, their dot product will be zero, confirming that they are perpendicular.

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

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Projections
Dot Product
Orthogonality

Formulas

Projection of A onto B: Proj_B(A) = (A • B / B • B) B
Orthogonal component of A with respect to B: A_perp = A - Proj_B(A)
Dot Product: A • B

Theorems

Orthogonal Projection Theorem
Dot Product and Perpendicularity

Suitable Grade Level

Undergraduate Linear Algebra