Math Problem Statement
Given the vectors u = <1, 2, 3> and v = <2, 1, 1>. Express u as the sum u = p + n, where p is parallel to v and n is orthogonal to v.
Solution
To express the vector as the sum , where is parallel to and is orthogonal to , we can follow these steps:
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Calculate the projection of onto to find :
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Calculate the dot products:
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Now calculate the projection:
So, .
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Calculate :
- Perform the subtraction:
Thus, the decomposition of is:
Summary:
- (parallel to )
- (orthogonal to )
Would you like further details or have any questions? Here are some follow-up questions to consider:
- How do you calculate the angle between two vectors?
- What are the geometric interpretations of parallel and orthogonal vectors?
- Can you derive the formula for the projection of one vector onto another?
- What properties do orthogonal vectors have in a vector space?
- How would you generalize this method for higher dimensions?
Tip: Always check that the dot product of and is zero to confirm orthogonality.
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Math Problem Analysis
Mathematical Concepts
Vector Decomposition
Linear Algebra
Formulas
Projection of u onto v: proj_v(u) = (u · v / v · v) * v
Orthogonal component: n = u - p
Theorems
Projection theorem
Suitable Grade Level
Grades 10-12
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