Math Problem Statement
In each part, find the vector component of ~v along ~b and the vector component of ~v orthogonal to ~b. Then sketch the vectors ~v, proj~b ~v, and ~v − proj~b ~v. (a) ~v = 2ˆi −ˆj, ~b = 3ˆi +ˆj
Solution
We are tasked with finding two vector components of with respect to :
- The vector component of along (denoted as ).
- The vector component of orthogonal to , which is .
Given:
Step 1: Calculate the projection of along
The formula for the projection of a vector along another vector is:
Find (dot product):
Find :
Now, calculate :
Step 2: Calculate the vector component of orthogonal to
This component is simply :
Step 3: Sketch the vectors
The vectors to be sketched are:
I'll prepare the sketch, which shows , , and the orthogonal component .
Would you like a more detailed explanation of the process?
Meanwhile, here are 5 questions to expand on the topic:
- How is the projection of a vector geometrically interpreted?
- What is the significance of the dot product in determining the projection?
- Can you explain why is orthogonal to ?
- How does the projection formula change when vectors are not in standard form?
- What are some real-world applications of vector projection?
Tip: The vector projection represents the "shadow" of one vector onto another, giving the component that aligns with the second vector.
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Math Problem Analysis
Mathematical Concepts
Vector Projection
Vector Orthogonality
Dot Product
Formulas
proj_b v = (v • b / b • b) b
v - proj_b v
Theorems
Projection Theorem
Orthogonality of Vector Components
Suitable Grade Level
Grades 10-12