Inner product, length, and orthogonality

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  1. Inner Product, Length, and Orthogonality Overview:
  2. The Length of a Vector
    • Also known as norm
    • Some properties with a length of a vector
    • Unit vectors
  3. Distance of two vectors/Orthogonal Vectors
    • Dist(u,v)=uv(u,v)=\lVert u-v \rVert
    • How to tell with two vectors are orthogonal
    • Orthogonal vectors with Pythagorean Theorem
  1. Utilizing the Inner Product
    Given that vector u and vector v, compute:
    uuvvv\frac{u \cdot u}{v \cdot v}v
    1. Given that vector u and vector v, compute:
      (uv)u\lVert (u \cdot v)u \rVert
      1. Finding the Unit vector
        Find the unit vector in the direction of the vector Finding the Unit vector
        1. Calculating the Distance
          Find the distance between the vectors Calculating the Distance, vector u and Calculating the Distance, vector v.
          1. Showing Orthogonality
            For what value(s) of bb make vectors uu and vv orthogonal if
            what value b make this two vectors orthogonal
            1. Proof Questions
              Show that the parallelogram is true for vectors uu and vv in R^n. In other words, show that:
              u+v2+uv2=2u2+2v2\lVert u+v \rVert^2 + \lVert u-v \rVert^2 = 2 \lVert u \rVert^2 + 2 \lVert v \rVert^2
              1. Suppose vector uu is orthogonal to vectors xx and yy. Show that uu is also orthogonal to x+yx+y.