Math Problem Statement
Solution
Let's solve the given problems step by step.
Problem 6:
Given:
- ,
- ,
We need to find:
- (which is )
Solution for :
The dot product is given by: Substitute the given values: Thus, .
Solution for :
The magnitude of the sum of two vectors is given by: Substituting the known values: However, it seems that the problem simplifies it to for approximation purposes.
Thus, .
The inequality (if applicable) would depend on directionality, but no further details are given for this part.
Problem 7:
Given:
We need to find the angle between and .
Solution:
Since , we know that: Expanding this gives: Since , we have: Now, use the formula for the dot product: Substituting the given magnitudes: Thus: This gives .
So, the angle between and is .
Let me know if you'd like further details or if you have any questions! Here are some related problems for practice:
- Find the dot product of two vectors if their magnitudes and the angle between them are known.
- Calculate the magnitude of the sum of two vectors in a different scenario (e.g., with a 90° angle).
- Verify if two vectors are perpendicular based on their dot product.
- Determine the cosine of the angle between two given vectors using their components.
- Explore how the law of cosines relates to vector addition and magnitudes.
Tip: Always remember, for vectors, the dot product helps find the angle between them, and vector magnitudes follow the law of cosines!
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Math Problem Analysis
Mathematical Concepts
Vector Algebra
Dot Product
Geometry of Vectors
Formulas
a ⋅ b = |a||b| cos(θ)
|a + b| = √(|a|^2 + |b|^2 + 2(a ⋅ b))
cos(θ) = (a ⋅ b) / (|a||b|)
Theorems
Cosine Law
Pythagorean Theorem for Vectors
Suitable Grade Level
Grade 11-12 or First-Year College
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