Inner product, length, and orthogonality

Inner product, length, and orthogonality


Let vectors uu and vv be:
vector u and vector v

Then the inner product of the two vectors will be:
inner product of vector u and vector v

Let u,vu,v and ww be vectors in Rn\Bbb{R}^n, and let cc be a scalar. Then,
1) uv=vuu \cdot v=v \cdot u
2) w(u+v)=wu+wv w(u+v)=w \cdot u+w \cdot v
3) (cu)v=c(uv)=u(cv) (cu) \cdot v=c(u \cdot v)=u \cdot (cv)
4) uu0u \cdot u \geq 0, and uu=0u \cdot u=0 only if u=0u=0

Suppose vector v. Then the length (or norm) of a vector vv is

v=v12+v22++vn2,v2=vv \lVert v \rVert = \sqrt{v_{1}^2+v_{2}^2+\cdots +v_{n}^2}\;,\; \lVert v \rVert ^2 = v \cdot v

Suppose dist(u,v)(u,v) is the distance between the vectors uu and vv. To find the distance between the two vectors, we calculate
dist(u,v)=uv(u,v)=\lVert u-v \rVert

If two vectors uu and vv are orthogonal to each other, then it must be true that
uv=0u \cdot v =0
  • Introduction
    Inner Product, Length, and Orthogonality Overview:
    The Inner Product
    • Transposing the first vector
    • Multiplying the two vectors
    • Multiplying the corresponding entries and adding them
    • Properties of Inner Product

    The Length of a Vector
    • Also known as norm
    • Some properties with a length of a vector
    • Unit vectors

    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

  • 2.
    Given that vector u and vector v, compute:
    (uv)u\lVert (u \cdot v)u \rVert

  • 3.
    Finding the Unit vector
    Find the unit vector in the direction of the vector Finding the Unit vector

  • 4.
    Calculating the Distance
    Find the distance between the vectors Calculating the Distance, vector u and Calculating the Distance, vector v.

  • 5.
    Showing Orthogonality
    For what value(s) of bb make vectors uu and vv orthogonal if
    what value b make this two vectors orthogonal

  • 6.
    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

  • 7.
    Suppose vector uu is orthogonal to vectors xx and yy. Show that uu is also orthogonal to x+yx+y.