MHB What is the Angle Between Vectors Using the Dot Product Formula?

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SUMMARY

The angle between the vectors \( v = -5\sqrt{3}i + 5j \) and \( w = 5i \) can be calculated using the dot product formula, defined as \( \cos{\theta} = \dfrac{\vec{v} \cdot \vec{w}}{|v| \, |w|} \). The dot product of two vectors in component form is computed as \( (a \vec{i} + b \vec{j}) \cdot (c \vec{i} + d \vec{j}) = ac + bd \). For the given vectors, the dot product results in a scalar quantity, which is essential for determining the angle between them.

PREREQUISITES
  • Understanding of vector notation and components
  • Familiarity with the dot product formula
  • Knowledge of trigonometric functions, specifically cosine
  • Ability to calculate magnitudes of vectors
NEXT STEPS
  • Practice calculating the dot product of different vector pairs
  • Learn how to derive angles between vectors using the cosine function
  • Explore applications of the dot product in physics and engineering
  • Study vector projections and their relationship to the dot product
USEFUL FOR

Students in mathematics, physics, and engineering, as well as anyone interested in vector analysis and geometric interpretations of angles between vectors.

brinlin
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Find the angle between the vectors $$v=-5\sqrt{3}i+5j$$ and $$w=5i$$
 
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I'd use the dot product formula ...

$\cos{\theta} = \dfrac{\vec{v} \cdot \vec{w}}{|v| \, |w|}$
 
when we use the dot product formula. What would we plug in for v and w.
 
to calculate the dot product of two vectors given in component form …

$(a \vec{i} + b \vec{j}) \cdot (c \vec{i} + d \vec{j}) = ac + bd$

… note the dot product is a scalar quantity
 
brinlin said:
when we use the dot product formula. What would we plug in for v and w.
? YOU said, in your first post that
$v= -5\sqrt{3}i+ 5j$
$w= 5i$.
 

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