Chapter 1 – Scalar and Vector Products

Resultant Force & Vector Operations

Overview of Chapter 1 Requirements

• Instructor’s request:

  • Produce a summary specifically about the “resulting force” material covered in Chapter 1.

  • Focus on two vector–operation tools used to evaluate the resulting force:

  1. Scalar (Dot) Product

  2. Vector (Cross) Product

• Reminder of alternate names:

  • Scalar product = dot product

  • Vector product = cross product

Scalar Product (Dot Product)

• Definition

  • Algebraic form (for vectors \mathbf{A} and \mathbf{B}):
    \mathbf{A}\cdot\mathbf{B}=|\mathbf{A}|\,|\mathbf{B}|\cos\theta

  • |\mathbf{A}| and |\mathbf{B}| = magnitudes

  • \theta = angle between the vectors

• Key properties

  • Produces a scalar (single numerical) result.

  • Commutative: \mathbf{A}\cdot\mathbf{B}=\mathbf{B}\cdot\mathbf{A}

• Significance in “resulting force” context

  • Determines how much of one force lies along the direction of another.

  • Useful for calculating work: W = \mathbf{F}\cdot\mathbf{s} (force along a displacement).

Vector Product (Cross Product)

• Definition

  • Algebraic form:
    \mathbf{A}\times\mathbf{B}=|\mathbf{A}|\,|\mathbf{B}|\sin\theta\,\mathbf{n}

  • \mathbf{n} = unit vector perpendicular to plane containing \mathbf{A} and \mathbf{B}, direction by right-hand rule

• Key properties

  • Produces a vector result (magnitude and direction).

  • Anti-commutative: \mathbf{A}\times\mathbf{B}= - (\mathbf{B}\times\mathbf{A})

• Significance in “resulting force” context

  • Determines a vector perpendicular to two force vectors; appears in torque: \boldsymbol\tau = \mathbf{r} \times \mathbf{F}.

Practical Example Links

• Dot product example (work): If |\mathbf{F}| = 10\,\text{N}, |\mathbf{s}| = 5\,\text{m}, \theta = 30^\circ, then W = 10\times5\times\cos30^\circ = 43.3\,\text{J}.

• Cross product example (torque): For a wrench of length 0.25\,\text{m} with force 40\,\text{N} applied at 90^\circ, |\boldsymbol\tau| = 0.25\times40\times\sin90^\circ = 10\,\text{N·m}; direction perpendicular to wrench plane.

Recap & Key Takeaways

• Resulting force analysis often decomposes into:

  • Projection (dot) to find parallel components.

  • Perpendicular influence or moment arm (cross) to find rotational effects.

• Memorizing both names—dot product for scalar, cross product for vector—ensures quick recall during problem-solving.