Math Problem Statement
Solution
The question asks to prove that the orthogonal projection of vector onto vector is perpendicular to the projection of the component of that is orthogonal to .
Let's break this down:
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Orthogonal projection of onto : The projection of onto , denoted as , is given by:
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Component of orthogonal to : The component of that is orthogonal to is given by: This is the part of that does not lie along , and it is perpendicular to .
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Proving perpendicularity: The dot product between the projection and the orthogonal component should be zero to prove they are perpendicular. We compute the dot product: Since is parallel to and is perpendicular to , their dot product will be zero, confirming that they are perpendicular.
Thus, the projection of onto is perpendicular to the orthogonal component of with respect to .
Would you like a deeper explanation or more details? Here are 5 related questions:
- What is the geometric significance of vector projections in linear algebra?
- How do we derive the formula for the orthogonal projection of one vector onto another?
- What is the relationship between dot product and perpendicularity of vectors?
- How can we use projections to decompose a vector into parallel and perpendicular components?
- 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
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