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Dot product

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Intros
Lessons
  1. Dot Product Overview:
  2. Dot Product and its Special Property
    • Multiplying the corresponding entries, and adding
    • Dot product = 0 →\to vectors are perpendicular
  3. Application to Dot Product
    • What is scalar projection? Vector projection
    • Formula for scalar projection: ∣v∣=a⋅b∣b∣|v| = \frac{a \cdot b}{|b|}
    • Formula for vector projection v=a⋅bb⋅bbv = \frac{a \cdot b}{b \cdot b}b
  4. More Properties of Dot Product
    • Order
    • Length
    • Distribution
    • Scalar
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Examples
Lessons
  1. Using the Dot Product
    Find the dot product of u=<−1,−2,7> u = <-1, -2 , 7> and v=<−2,1,−2>v = <-2,1,-2> .
    1. Find the dot product of u=<1,−5,−3>u= <1, -5, -3> and v=<−1,1,2>v= <-1, 1, 2>.
      1. Using the Dot Product Property
        Suppose two vectors u=<a,4,−3> u = < a, 4, -3> and v=<1,2,3> v=<1, 2, 3> are perpendicular. Find aa.
        1. Finding Scalar and Vector Projections
          Find the scalar and vector projection of BA⃗\vec{BA} onto CA⃗\vec{CA} if A=(1,0,2)A=(1, 0, 2), B=(3,−2,1)B=(3, -2, 1) and C=(−4,1,5)C=(-4, 1, 5).
          1. Verifying Properties of Dot Product
            Use the two vectors u=<3,1,5>u=<3, 1, 5> and v=<1,4,−6>v=<1,4,-6> to show that:

            u⋅u=∣u∣2u \cdot u = |u|^2

            1. Use the 3 vectors u=<3,1,5>u=<3, 1, 5>, v=<1,4,−6>v=<1,4,-6>, and w=<1,0,3>w=<1, 0, 3> to show that:

              u⋅(v+w)=u⋅v+u⋅wu \cdot (v+w)=u \cdot v+ u \cdot w

              Topic Notes
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              Notes:

              Dot Product
              Let u=<a,b,c>u=<a,b,c> and v=<d,e,f>v=<d,e,f>. Then the dot product of these two vectors will be:

              u⋅v=ad+be+dfu \cdot v = ad + be + df

              Dot Product Property
              If the dot product of two vectors uu and vv gives 0, then the vectors are perpendicular. In other words,

              u⋅v=0→u \cdot v=0 \to perpendicular vectors

              Scalar and Vector Projection
              Suppose we have two vectors aa and bb. Suppose they create an angle θ\theta such that we get the following picture:
              projection

              ∣v∣=a⋅b∣b∣ |v| = \frac{a \cdot b } {|b|}

              To find the vector projection aa onto bb (which is v), we use the formula:

              v=a⋅bb⋅bb v = \frac{a \cdot b}{b \cdot b}b

              Additional Dot Product Properties
              Let u,v,wu, v, w be vectors and cc be a scalar. Then the properties of dot products are:
              1. u⋅u=∣u∣2u \cdot u = |u|^2
              2. u⋅v=v⋅uu \cdot v = v \cdot u
              3. u⋅(v+w)=u⋅v+u⋅wu \cdot (v+w) = u \cdot v + u \cdot w
              4. (cu)⋅v=u⋅(cv)=c(u⋅v) (cu) \cdot v = u \cdot (cv) = c(u \cdot v)