Math Problem Statement
Solution
We are given the following information:
We are asked to find .
Step-by-step solution:
-
Formula for the magnitude of the vector sum:
Expand the dot product:
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Substitute known values:
Now substitute these into the expanded expression: Simplifying:
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Take the square root:
Thus, the magnitude of is approximately 13.34.
Would you like further details or have any other questions?
Here are 5 related questions:
- How do we generalize the formula for the magnitude of the sum of two vectors?
- Can you explain how dot products are used to calculate angles between vectors?
- How does scaling a vector by a scalar affect its magnitude?
- How would the calculation change if and were orthogonal?
- What is the significance of the dot product being positive, negative, or zero?
Tip: When dealing with vector sums, always expand using the distributive property and remember that gives crucial information about the angle between the vectors.
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Math Problem Analysis
Mathematical Concepts
Vectors
Vector Magnitude
Dot Product
Formulas
Magnitude of a vector: ||a|| = sqrt(a · a)
Dot product formula: a · b = ||a|| ||b|| cos(θ)
Magnitude of the sum of vectors: ||u + 3v|| = sqrt((u + 3v) · (u + 3v))
Theorems
Pythagorean Theorem in Vector Form
Suitable Grade Level
Grades 11-12
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