Understanding Dot Product: Identity and Unit Vectors

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SUMMARY

The dot product of a unit vector (denoted as \(\hat{e}\)) with a non-unit vector (\(\vec{A}\)) is calculated using the formula \(\hat{e} \cdot \vec{A} = |\vec{A}| \cos \theta\), where \(\theta\) is the angle between the two vectors. This relationship indicates that the dot product represents the component of the non-unit vector along the direction of the unit vector. Without additional information about the angle or magnitude of \(\vec{A}\), this is the most simplified form of the expression.

PREREQUISITES
  • Understanding of vector notation and operations
  • Familiarity with trigonometric functions, specifically cosine
  • Knowledge of unit vectors and their properties
  • Basic grasp of geometric interpretations of dot products
NEXT STEPS
  • Study the geometric interpretation of dot products in vector analysis
  • Learn about the properties of unit vectors in three-dimensional space
  • Explore applications of dot products in physics, particularly in work and energy calculations
  • Investigate the relationship between dot products and projections of vectors
USEFUL FOR

Students of mathematics and physics, educators teaching vector calculus, and professionals in engineering fields who require a solid understanding of vector operations.

don_anon25
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What is the dot product of a unit vector (e) with a non-unit vector (A)? Is this some sort of identity?

Thanks,
don_anon25
 
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don_anon25 said:
What is the dot product of a unit vector (e) with a non-unit vector (A)? Is this some sort of identity?

Thanks,
don_anon25

Let a be a unit vector and b be another vector. a * b = |a||b|cosθ. Since a is a unit vector, this becomes a * b = |b|cosθ. I believe that without more information, this is as far as it can be simplified.
 
don_anon25 said:
What is the dot product of a unit vector (e) with a non-unit vector (A)? Is this some sort of identity?

Thanks,
don_anon25

<br /> \hat e\cdot\vec A is the component of \vec A along the \hat e-direction.
 

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