Vector Dot Products

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Vector Dot Products

• AP Physics C
• L. Alewine

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Unit Vectors

• A vector of length 1 is a unit vector.
• Example

Vector A has magnitude and direction.
Unit Vector
Divide by the length (magnitude) of vector A.
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Magnitude Length

• How do you find the length of a vector at some
  angle ? ?
• When there were only two components (x and y) we
  used the Pythagorean theorem.
• To accommodate more dimensions, we use the
  distance modulus.

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Distance Modulus
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Common Unit Vectors
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Dot or Scalar Product
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What IS a dot product?

• The dot product has the geometric interpretation
  as the length of the projection of X onto the
  unit vector Y when the two vectors are placed so
  that their tails coincide.

• Think of the projection of X as the component of
  the vector X in the unit vector vector Ys
  direction.

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Write vectors in terms of rectangular components.
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Calculate the Dot Product
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Derivatives

• The derivative of a dot product of vectors is

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Think about it

• What is the angle between the vectors A and B if

The angle must be 180 because the cosine of 180
is -1!
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Think about it some more

• What is the angle between the vectors A and B if

The angle must be 90 because the cosine of 90
is 0! These vectors are known to be orthogonal.
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The Angle Between 2 Vectors
Calculate the dot product of the two vectors.
Then divide by the product of their lengths.
Use the distance modulus to determine the length
of each vector.
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How its done

• Example
• Find the work done against gravity to move a 10
  kg baby from the point (2,3) to the point (5,7)? 

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Solution

• We have that the force vector is 
•         F    ma    (10)(-9.8j)    -98j
• and the displacement vector is
•         d    (5 - 2) i    (7 - 3) j    3i 4j
  
• The work is the dot product
•         W    F . d    (-98j) . (3i 4j)  
•           (0)(3) (-98)(4)    -392
• Notice the negative sign verifies that the work
  is done against gravity.  Hence, it takes 392 J
  of work to move the baby.