Section 12.3: The Dot Product of Two Vectors

Welcome to Professor Baker's Math Class! Today, we're diving into Section 12.3, focusing on the dot product of vectors. The dot product is a fundamental operation with many applications, so let's get started!

Definition of the Dot Product

To find the dot product of vectors a and b, we multiply corresponding components and then add the results. Mathematically, if we have two vectors:

a = $(a_1, a_2, a_3)$ and b = $(b_1, b_2, b_3)$

Then, the dot product of a and b, denoted as a · b, is given by:

$$a · b = a_1b_1 + a_2b_2 + a_3b_3$$

Let's look at some examples:

  • (2, 4) · (3, -1) = (2)(3) + (4)(-1) = 6 - 4 = 2
  • (-1, 7, 4) · (6, 2, -$\frac{1}{2}$) = (-1)(6) + (7)(2) + (4)(-$$\frac{1}{2}$) = -6 + 14 - 2 = 6
  • (i + 2j - 3k) · (2j - k) = (1)(0) + 2(2) + (-3)(-1) = 0 + 4 + 3 = 7

Properties of the Dot Product

If a, b, and c are vectors in V3 (3-dimensional space) and c is a scalar, then the dot product has the following properties:

  1. a · a = |a|$^2$ (The dot product of a vector with itself is the square of its magnitude)
  2. a · b = b · a (Commutative property)
  3. a · (b + c) = a · b + a · c (Distributive property)
  4. (ca) · b = c(a · b) = a · (cb) (Scalar multiplication)
  5. 0 · a = 0 (The dot product of the zero vector with any vector is zero)

Angle Between Vectors

The dot product can be used to find the angle θ between two vectors a and b. The relationship is given by:

$$a · b = |a||b| cos θ$$

Therefore, the cosine of the angle between the vectors is:

$$cos θ = \frac{a · b}{|a||b|}$$

And the angle θ can be found using the inverse cosine function:

$$θ = arccos(\frac{a · b}{|a||b|})$$

For example: If the vectors a and b have lengths 4 and 6, and the angle between them is $\frac{π}{3}$, then

$$a · b = (4)(6)cos(\frac{π}{3}) = 24 * (\frac{1}{2}) = 12$$

Perpendicular Vectors

Two vectors are perpendicular (orthogonal) if their dot product is zero:

a · b = 0

If a · b > 0, the angle between the vectors is acute (less than 90°). If a · b < 0, the angle between the vectors is obtuse (greater than 90°).

Direction Cosines and Direction Angles

The direction angles α, β, and γ are the angles that a vector a makes with the positive x, y, and z axes, respectively. The cosines of these angles are called direction cosines.

If a = $$, then:

  • cos α = $\frac{a_1}{|a|}$
  • cos β = $\frac{a_2}{|a|}$
  • cos γ = $\frac{a_3}{|a|}$

And a = |a|(cos α, cos β, cos γ)

Work Done by a Constant Force

The work W done by a constant force F in moving an object a distance D is given by the dot product:

$$W = F · D = |F||D| cos θ$$

Where θ is the angle between the force and the direction of motion.

Great job working through the dot product! Keep practicing, and you'll master these concepts in no time!