Scalars and Vectors

Scalars and Vectors

• Scalars are quantities defined by a positive or negative number, examples include length, area, volume, mass, time, and temperature.
• Vectors are quantities possessing magnitude, direction, and sense.
• Vectors are represented graphically by an arrow.
  • The arrow's length indicates magnitude.
  • The angle relative to a fixed axis indicates direction.
  • The arrowhead indicates the sense.
• F denotes a force vector and F or |F| signifies its magnitude.

Vector Operations

• The product of a vector A and a scalar a is aA.
  • The magnitude |aA| equals |a| |A|.
  • The sense is identical to A if a > 0 but opposite if a < 0.
• Vector quantities follow the parallelogram law of addition: R = A + B.
• Vector addition is commutative: R = A + B = B + A (Triangle Rule).
• Subtraction is a special case of addition: R' = A - B = A + (-B).
• Vectors can be added in any order: R = (A + B) + C = A + (B + C).
• The resultant vector R from multiple forces is R = ΣF = F₁ + F₂ + F₃.

Vector Addition of Forces

• Determining the resultant force involves graphical construction (less accurate) or trigonometry (more accurate).

Procedure for Analysis

• Determine angles between vectors and parallelogram sides.
• Use the Law of Cosines to calculate the resultant force magnitude: $R = \sqrt{A^2 + B^2 - 2AB \cos(c)}$.
• Use the Law of Sines to determine the angle of the resultant force: $\frac{A}{\sin(a)} = \frac{B}{\sin(b)} = \frac{R}{\sin(c)}$.

Cartesian Vectors

• A right-handed coordinate system uses orthogonal axes (x, y, z) for spatial orientation; thumb along x, fingers curl from y to z.
• A vector A in Cartesian form: A = $A_x$i + $A_y$j + $A_z$k.
  • $A_x$, $A_y$, $A_z$ represent scalar components along the x, y, z axes.
  • $A_x$i, $A_y$j, $A_z$k represent vector components along the x, y, z axes.
• Magnitude of a Cartesian vector: $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$.
• Direction defined by angles α, β, γ; their cosines are direction cosines:
  • $\cos(\alpha) = \frac{A_x}{A}$, $\cos(\beta) = \frac{A_y}{A}$, $\cos(\gamma) = \frac{A_z}{A}$.
• Relationship between direction cosines: $\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1$.
• A unit vector u$_A$ has a magnitude of 1 and indicates direction:
  • u$_A$ = $\frac{\mathbf{A}}{A}$ = $\frac{A_x}{A}$i + $\frac{A_y}{A}$j + $\frac{A_z}{A}$k = $\cos(\alpha)$i + $\cos(\beta)$j + $\cos(\gamma)$k.
  • Therefore, $\cos(\alpha)$ = $u_{Ax}$, $\cos(\beta)$ = $u_{Ay}$, and $\cos(\gamma)$ = $u_{Az}$.

Addition of Cartesian Vectors

• Concurrent force systems involve multiple forces acting at one point.
• The resultant force F$_R$ is the sum of all forces:
  • F$_R$ = ΣF = Σ$F_x$i + Σ$F_y$j + Σ$F_z$k.
  • $F_{Rx}$ = Σ$F_x$, $F_{Ry}$ = Σ$F_y$, $F_{Rz}$ = Σ$F_z$.
  • $F_R$ = $\sqrt{F_{Rx}^2 + F_{Ry}^2 + F_{Rz}^2}$.
  • $\cos(\alpha) = \frac{F_{Rx}}{F_R}$, $\cos(\beta) = \frac{F_{Ry}}{F_R}$, $\cos(\gamma) = \frac{F_{Rz}}{F_R}$.

Position Vectors

• Position vector r locates a point in space relative to another.
  • r = xi + yj + z*k.
• Magnitude of r: $r = \sqrt{x^2 + y^2 + z^2}$.
• A force vector F directed along a line:
  • F = $F$u = $F \frac{\mathbf{r}}{r}$ = $F \frac{(x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j} + (z_2 - z_1)\mathbf{k}}{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}$.
  • F = $F_x$i + $F_y$j + $F_z$k.

Dot Product

• A ⋅ B = $A B \cos(\theta)$, where θ is the angle ($0 ≤ \theta ≤ 180°$) between vectors A and B.

Laws of Operation

• Commutative Law: A ⋅ B = B ⋅ A.
• Scalar Multiplication: a (A ⋅ B) = (aA) ⋅ B = A ⋅ (aB).
• Distributive Law: A ⋅ (B + C) = (A ⋅ B) + (A ⋅ C).

Cartesian Vector Formulation for Dot Product

• A ⋅ B = ($A_x$i + $A_y$j + $A_z$k) ⋅ ($B_x$i + $B_y$j + $B_z$k).
• A ⋅ B = $A_x B_x + A_y B_y + A_z B_z$.

Applications of Dot Product

• Determining the angle between two vectors:
  • $\cos(\theta) = \frac{\mathbf{A} \cdot \mathbf{B}}{AB} = \frac{A_x B_x + A_y B_y + A_z B_z}{\sqrt{A_x^2 + A_y^2 + A_z^2} \sqrt{B_x^2 + B_y^2 + B_z^2}}$.
• Finding a vector component in a specific direction:
  • $A' = A \cos(\theta) = \mathbf{A} \cdot \mathbf{u}$.
  • A' = (A ⋅ u) u.