12.3: Vectors and the Dot Product

Vectors and the Dot Product

• The dot product (scalar product) is an operation to multiply two vectors, returning a real number.

• Defined as: $a \cdot b = a<em>1 b</em>1 + a<em>2 b</em>2 + a<em>3 b</em>3$ for vectors in $\mathbb{R}^3$.

Properties of the Dot Product

• Symmetry: $a \cdot b = b \cdot a$

• Distributive: $a \cdot (b + c) = a \cdot b + a \cdot c$

• Scalar multiplication: $(c \cdot a) \cdot b = c (a \cdot b)$

• Zero vector property: $a \cdot 0 = 0$

Geometric Interpretation

• Given angle $\theta$ between vectors $a$ and $b$:

  • $a \cdot b = |a||b| \cos(\theta)$

  • If $\theta = 0$: vectors are parallel; $\theta = \pi$: vectors are opposite.

Angle Calculation

• To find angle between two vectors:

  • $\cos(\theta) = \frac{a \cdot b}{|a||b|}$

Orthogonality

• Two nonzero vectors are orthogonal if $a \cdot b = 0$.

Direction Angles and Cosines

• Direction angles $\alpha, \beta, \gamma$ are angles with axes; cosines are direction cosines:

  • $\cos \alpha = \frac{a<em>1}{|a|}, \cos \beta = \frac{a</em>2}{|a|}, \cos \gamma = \frac{a_3}{|a|}$

  • Must satisfy: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$

Projections

• Vector projection of $b$ onto $a$: $\text{proj}_a b = \frac{a \cdot b}{|a|^2} a$

• Scalar projection (component): $\text{comp}_a b = |b| \cos(\theta)$.

Work Application

• Work done by force $F$ over distance $D$:

  • $W = F \cdot D$

  • Use angle between force and displacement vectors to find the work done efficiently.