Scalars, Vectors, and Vector Operations

Scalars and Vectors

• A scalar is defined by magnitude alone.
• A vector is defined by magnitude and direction.

Vector Algebra

Vector Addition

• Vectors are added geometrically.
• Triangle rule and parallelogram rule are used.
  • Commutative law: $\mathbf{P} + \mathbf{Q} = \mathbf{Q} + \mathbf{P}$
  • Associative law: $\mathbf{P} + (\mathbf{Q} + \mathbf{R}) = (\mathbf{P} + \mathbf{Q}) + \mathbf{R}$

Vector Subtraction

• $\mathbf{P} - \mathbf{Q} = \mathbf{P} + (-\mathbf{Q})$

Scalar Multiplication

• $\mathbf{P} = a\mathbf{Q}$

Unit Vectors, Cartesian Components, and Direction Cosines

Unit Vectors

• A unit vector has a magnitude of one.
• $\mathbf{U} = \frac{\mathbf{A}}{A}$
  • $\mathbf{U}$ represents the unit vector of $\mathbf{A}$.
  • $\mathbf{A}$ represents the vector.
  • $A$ represents the magnitude of the vector $\mathbf{A}$.

Cartesian Components

• $\mathbf{A} = A_x\mathbf{i} + A_y\mathbf{j} + A_z\mathbf{k}$
  • $A_x\mathbf{i}$, $A_y\mathbf{j}$, $A_z\mathbf{k}$ are vector components of $\mathbf{A}$ in the x, y, and z directions.
  • $A_x$, $A_y$, $A_z$ are scalar components of $\mathbf{A}$ in the x, y, and z directions.
• Magnitude of $\mathbf{A}$: $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$

Addition and Subtraction of Cartesian Vectors

• $\mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x)\mathbf{i} + (A_y + B_y)\mathbf{j} + (A_z + B_z)\mathbf{k}$
• $\mathbf{R} = \mathbf{A} - \mathbf{B} = (A_x - B_x)\mathbf{i} + (A_y - B_y)\mathbf{j} + (A_z - B_z)\mathbf{k}$

Direction Cosines

• Direction cosines are the cosines of the angles between a vector and the coordinate axes.
• $\cos\theta_x = \frac{A_x}{A}$, $\cos\theta_y = \frac{A_y}{A}$, $\cos\theta_z = \frac{A_z}{A}$
  • $\theta_x$, $\theta_y$, $\theta_z$ are the direction angles of $\mathbf{A}$.
  • $\cos^2\theta_x + \cos^2\theta_y + \cos^2\theta_z = 1$
  • $\mathbf{U} = \cos\theta_x\mathbf{i} + \cos\theta_y\mathbf{j} + \cos\theta_z\mathbf{k}$

Dot Product

• $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos\theta = AB\cos\theta$
  • Commutative law: $\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$
  • Scalar multiplication: $a(\mathbf{A} \cdot \mathbf{B}) = (a\mathbf{A}) \cdot \mathbf{B} = \mathbf{A} \cdot (a\mathbf{B}) = (\mathbf{A} \cdot \mathbf{B})a$
  • Distributive law: $\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}$
• Cartesian Vector Formulation: $\mathbf{A} \cdot \mathbf{B} = (A_x\mathbf{i} + A_y\mathbf{j} + A_z\mathbf{k}) \cdot (B_x\mathbf{i} + B_y\mathbf{j} + B_z\mathbf{k}) = A_xB_x + A_yB_y + A_zB_z$

Applications

• The angle between two vectors: $\cos\theta = \frac{\mathbf{A} \cdot \mathbf{B}}{AB} = \frac{A_xB_x + A_yB_y + A_zB_z}{AB}$
• The component of a vector parallel to another vector: $A_{\parallel} = A\cos\theta = \mathbf{A} \cdot \mathbf{U}_B$

Cross Product

• $\mathbf{C} = \mathbf{A} \times \mathbf{B} = AB\sin\theta \mathbf{U}$

  • $\theta$ is the angle between the tails of $\mathbf{A}$ and $\mathbf{B}$ ($0^\circ \leq \theta \leq 180^\circ$).
  • $\mathbf{U}$ is a unit vector perpendicular to both $\mathbf{A}$ and $\mathbf{B}$.
  • Commutative law: $\mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A})$
  • Scalar multiplication: $a(\mathbf{A} \times \mathbf{B}) = (a\mathbf{A}) \times \mathbf{B} = \mathbf{A} \times (a\mathbf{B}) = (\mathbf{A} \times \mathbf{B})a$
  • Distributive law: $\mathbf{A} \times (\mathbf{B} + \mathbf{C}) = \mathbf{A} \times \mathbf{B} + \mathbf{A} \times \mathbf{C}$

• Cartesian Vector Formulation: $\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix} = (A_yB_z - A_zB_y)\mathbf{i} - (A_xB_z - A_zB_x)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k}$

Applications

• Moment of a force: $\mathbf{M}_O = \mathbf{r} \times \mathbf{F}$