Notes on Section 1.5: Dot Product, Norm, and Angle

Section 1.5: Dot Product, Norm, and Angle

• Context: This topic presents the simplest algebraic formula involving two vectors and shows how such a bilinear form is used to derive geometric quantities like lengths, distances, and angles.

• Key takeaway: The dot product is a bilinear, symmetric form that maps two vectors to a real number and encodes length and angle information.

• Dot product basics

  • Definition (in R^n): for vectors x, y,
    $x \cdot y = \sum<em>{i=1}^n x</em>i y_i.$
  • Commutativity: $x \cdot y = y \cdot x.$
  • Observations about bilinearity:
  • If you fix one vector, the map in the other variable is linear.
  • If you fix y and vary x, the function f(x) = x·y is linear in x.
  • If you fix x and vary y, the function g(y) = x·y is linear in y.
  • A note on notation: there is a linear map interpretation when one argument is fixed, commonly denoted by something like Tx(y) = x·y or Ty(x) = x·y. The general dot product is bilinear in its two arguments.
  • In context of inner products, the dot product is a prototype example of a bilinear, symmetric, positive-definite form on a real vector space.

• The special case x·x and the norm

  • Definition of the length (norm) of a vector x:
    $|x| = \sqrt{x \cdot x} = \sqrt{\sum<em>{i=1}^n x</em>i^2}.$
  • Distance from origin to tip of x equals the length: the geometric interpretation of the norm.
  • Length squared: $x \cdot x = |x|^2.$

• Distance between two vectors and the law of cosines

  • Distance between x and y (in Euclidean space) is
    $d(x, y) = |x - y| = \sqrt{(x<em>1 - y</em>1)^2 + \cdots + (x<em>n - y</em>n)^2}.$
  • Expressing the distance using dot product:
    $|x - y|^2 = (x - y) \cdot (x - y) = |x|^2 + |y|^2 - 2 x \cdot y.$
  • This identity arises from expanding the dot product and using linearity and symmetry.
  • Law of cosines connection: for vectors x and y with angle θ between them,
    $|x - y|^2 = |x|^2 + |y|^2 - 2 |x| |y| \cos\theta.$
  • Consequently, the dot product encodes the angle via
    $x \cdot y = |x| |y| \cos\theta.$
  • Therefore, by dividing by the product of the norms (when both are nonzero), the cosine of the angle is
    $\cos\theta = \frac{x \cdot y}{|x| \|y|}.$

• Interpreting the angle and unit vectors

  • The dot product becomes a measure of alignment: if x and y point in the same direction, x·y is large and positive; if orthogonal, x·y = 0; if opposite, x·y is negative.
  • In the plane, this is often illustrated by a right triangle or by projecting one vector onto another and using the Pythagorean relation.
  • If either vector is zero, the angle is not defined in the usual sense, and the cosine formula is not applicable due to division by zero in the denominator.

• Coordinate view and examples

  • In R^2, with x = (x1, x2) and y = (y1, y2):
    $x \cdot y = x1 y1 + x2 y2,$
    $|x| = \sqrt{x1^2 + x2^2}, \quad |y| = \sqrt{y1^2 + y2^2}.$
  • Distance example in coordinates:
    $d(x, y) = \sqrt{(x1 - y1)^2 + (x2 - y2)^2}.$
  • Angle example with standard basis in R^2: for e1 = (1,0) and e2 = (0,1),
    $e1 \cdot e2 = 0, \quad |e1| = |e2| = 1, \quad \cos\theta = 0, \quad \theta = \frac{\pi}{2}.$

• Practical identities and derivations

  • Expand (y - x) in a bilinear context to illustrate linearity in each argument:
  • For fixed u, v, and varying y, x, one can derive that
    $(y - x) \cdot u = y \cdot u - x \cdot u,$
    and similarly for another fixed vector v if needed. The transcript notes the use of bilinearity to decompose such expressions.
  • Substituting x = y leads to the length relation:
    $|x|^2 = x \cdot x.$

• Conceptual recap and significance

  • The dot product ties together algebra and geometry: it provides a coordinate-free way to measure lengths and angles, and it underpins projections, distances, and many geometric constructions.
  • It also serves as a basic axiom for inner products: the bilinearity, symmetry, and positive definiteness encode the geometry of Euclidean space.
  • The norm is defined in terms of the dot product, and all distance and angle notions derive from the norm and the dot product.

• Quick reference formulas (LaTeX)

  • Dot product: $x \cdot y = \sum<em>{i=1}^n x</em>i y_i$
  • Norm: $|x| = \sqrt{x \cdot x} = \sqrt{\sum<em>{i=1}^n x</em>i^2}$
  • Distance: $d(x, y) = |x - y| = \sqrt{(x<em>1 - y</em>1)^2 + \cdots + (x<em>n - y</em>n)^2} = \sqrt{(x - y) \cdot (x - y)}$
  • Law of cosines form: $|x - y|^2 = |x|^2 + |y|^2 - 2 |x| |y| \cos\theta$
  • Dot product in terms of angle: $x \cdot y = |x| |y| \cos\theta$
  • Cosine of angle: $\cos\theta = \frac{x \cdot y}{|x| |y|}$ (provided |x| > 0 and |y| > 0)

• Additional context from the lecture notes embedded in the transcript

  • The discussion reflects a typical progression: define a simple bilinear form (the dot product), observe linearity in each argument separately, discuss the norm and distance, and connect to the angle via the law of cosines.
  • The speaker notes a common teaching anecdote about notation versus thinking versus writing, illustrating the potential mismatch between what is thought and what is written on the board.

• Practical implications and applications

  • Determines whether two vectors are at right angles (orthogonality) via x·y = 0.
  • Enables projection of one vector onto another: proj_y(x) = (x·y / y·y) y.
  • Provides a natural notion of distance and similarity between vectors in Euclidean space, which underpins many algorithms in data analysis, computer graphics, physics, and engineering.

• Summary clause

  • The dot product is a foundational tool that simultaneously encodes magnitude and angular information, linking algebraic operations with geometric interpretation through the norm and cosine of the angle between vectors.