Vectors and Coordinate Systems

Vectors and Coordinate Systems

Vectors (3.1)

• A vector quantity includes both magnitude (always positive) and direction.
• Examples of vector quantities include displacement and velocity.
• An arrow represents a vector, with the tail indicating the starting point or point of application of the vector quantity.

Displacement Vectors (3.2)

• A displacement is defined as the straight-line connection from an initial position to a final position.
• Its magnitude is the straight-line distance between these two positions.
• Examples:
  • $\vec{d<em>1} = (30 \text{ m, east})$ where the magnitude $d</em>1 = 30 \text{ m}$.
  • $\vec{d<em>2} = (20 \text{ m, } 40^\circ \text{ north of west})$ where the magnitude $d</em>2 = 20 \text{ m}$.

Identical Vectors (3.2)

• Two vectors are considered equal or identical if they possess the same magnitude and the same direction.
• The specific starting and ending points of the arrows representing the vectors do not affect their equality as long as their magnitude and direction are identical.
• Example: Two displacement vectors from paths that start at different locations, end at different locations, and have different shapes can still be equivalent if their overall straight-line displacement (magnitude and direction) is the same: $\vec{d<em>1} = \vec{d</em>2}$.

Vector Addition (3.2)

• Graphical Addition - Head-to-Tail Method:
  • Successive displacements or any two vectors can be added graphically by placing them head-to-tail.
  • The resultant vector (or overall displacement) is then drawn from the free tail of the first vector to the free head of the last vector.
  • For two vectors, $\vec{d}<em>{net} = \vec{d</em>1} + \vec{d_2}$.
• Commutative Property:
  • Vector addition is commutative, meaning the order in which vectors are added does not change the resultant vector:
    $\vec{A} + \vec{B} = \vec{B} + \vec{A}$
• Using Geometry to Find Magnitude and Direction:
  • When adding perpendicular vectors (e.g., east and north), geometry (Pythagorean theorem and trigonometry) can be used.
  • Example: Adding $\vec{A} = (40 \text{ m, east})$ and $\vec{B} = (30 \text{ m, north})$.
    • Magnitude of resultant $\vec{C}$ ($\vec{C} = \vec{A} + \vec{B}$): $C = \sqrt{A^2 + B^2} = \sqrt{(40 \text{ m})^2 + (30 \text{ m})^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50 \text{ m}$.
    • Direction of resultant $\vec{C}$ ($\theta$ relative to the east direction):
      $\theta = \tan^{-1}\left(\frac{B}{A}\right) = \tan^{-1}\left(\frac{30 \text{ m}}{40 \text{ m}}\right) = 36.9^\circ$ (north of east).
    • Resultant vector: $\vec{C} = (50 \text{ m, } 36.9^\circ \text{ north of east})$.
• Adding Three or More Vectors:
  • The head-to-tail method extends to adding any number of vectors.
  • The general formula is $\vec{d}<em>{net} = \vec{d</em>1} + \vec{d<em>2} + \vec{d</em>3} + \vec{d_4} + \dots$.
  • The resultant vector connects the tail of the first vector to the head of the last vector.

Multiplication By a Scalar (3.2)

• Multiplying by a Positive Scalar ($c > 0$):
  • Results in a new vector with a different magnitude but the same direction.
  • If $\vec{B} = c\vec{A}$ where $c$ is a positive scalar, then the magnitude $B = cA$ and the direction $\theta<em>B = \theta</em>A$.
  • This principle explains why, in equations like $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$, if $\Delta t$ is positive, the acceleration vector $\vec{a}$ points in the same direction as the change in velocity vector $\Delta \vec{v}$.
• Multiplying by a Negative Scalar (e.g., -1):
  • Multiplying a vector by -1 (or any negative scalar) reverses its direction while leaving its magnitude unchanged.
  • If $\vec{A}$ has direction $\theta<em>A$, then $-\vec{A}$ has direction $\theta</em>A + 180^\circ$.

Vector Subtraction (3.2)

• To subtract one vector ($\vec{B}$) from another vector ($\vec{A}$):
  1. Find the negative of the vector to be subtracted ($-\vec{B}$).
  2. Add this negative vector to the first vector using the head-to-tail method.
• The operation is expressed as $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$.
• Important Note: Vector subtraction is not commutative; $\vec{A} - \vec{B} \neq \vec{B} - \vec{A}$. In fact, $\vec{A} - \vec{B} = -(\vec{B} - \vec{A})$.

Component Vectors (3.3)

• Any vector can be decomposed or resolved into component vectors that are parallel to the coordinate axes.
• These component vectors add up to give the original vector (e.g., $\vec{A} = \vec{A<em>x} + \vec{A</em>y}$).
• A component of a vector is the magnitude of the corresponding component vector, accompanied by a sign to indicate its direction along the axis.
• Graphically, components can be found by drawing a rectangle around the vector with sides parallel to the x and y axes.
• Example: For vector $\vec{A}$, its components are $A<em>x$ and $A</em>y$.

Trigonometry and Vectors (3.3)

• Calculating components from a vector's magnitude ($A$) and direction ($\theta$):
  • The x-component: $A_x = A \cos \theta$
  • The y-component: $A_y = A \sin \theta$
• Calculating a vector's magnitude and direction from its components ($A<em>x$, $A</em>y$):
  • Magnitude: $A = \sqrt{A<em>x^2 + A</em>y^2}$ (Pythagorean theorem)
  • Direction: $\theta = \tan^{-1}\left(\frac{A<em>y}{A</em>x}\right)$
• Crucial Caveats:
  • These formulas ($\tan^{-1}$ specifically) are not universally valid for every vector due to the nature of the inverse tangent function, which typically returns angles in a limited range. The quadrant of the vector must be considered to determine the correct direction.
  • Always ensure your calculator is set to degrees mode when performing these calculations.

Vector Algebra: Unit Vectors (3.4)

• Unit vectors provide a convenient way to express the components of any vector.
• Definitions of Cartesian Unit Vectors:
  • $\hat{i} \equiv (1, \text{ positive x-direction})$
  • $\hat{j} \equiv (1, \text{ positive y-direction})$
  • $\hat{k} \equiv (1, \text{ positive z-direction})$
• Representing a Vector with Unit Vectors:
  • A vector $\vec{A}$ can be written as the sum of its component vectors:
    $\vec{A} = \vec{A<em>x} + \vec{A</em>y} + \vec{A_z}$
  • Using unit vectors, this becomes:
    $\vec{A} = A<em>x\hat{i} + A</em>y\hat{j} + A_z\hat{k}$
  • Example: $\vec{A} = (2\hat{i} - 1\hat{j} + 0\hat{k})\text{m} = (2\hat{i} - \hat{j})\text{m}$.

Algebraic Vector Addition (3.4)

• When using unit vectors, they can be treated as algebraic variables.
• Method: To add vectors (e.g., $\vec{D} = \vec{A} + \vec{B} + \vec{C}$), express each vector in its component form, then group coefficients of the same unit vector.
  • If $\vec{A} = A<em>x\hat{i} + A</em>y\hat{j}$, $\vec{B} = B<em>x\hat{i} + B</em>y\hat{j}$, and $\vec{C} = C<em>x\hat{i} + C</em>y\hat{j}$:
  • $\vec{D} = (A<em>x\hat{i} + A</em>y\hat{j}) + (B<em>x\hat{i} + B</em>y\hat{j}) + (C<em>x\hat{i} + C</em>y\hat{j})$
  • $\vec{D} = (A<em>x + B</em>x + C<em>x)\hat{i} + (A</em>y + B<em>y + C</em>y)\hat{j}$
• This implies that the components of the resultant vector are the sums of the corresponding components of the individual vectors:
  • $D<em>x = A</em>x + B<em>x + C</em>x$
  • $D<em>y = A</em>y + B<em>y + C</em>y$
• This generalizes to any number of vectors and to three dimensions (including $D<em>z = A</em>z + B<em>z + C</em>z$).

Algebraic Vector Subtraction and Scalar Multiplication (3.4)

• Algebraic Vector Subtraction:
  • If $\vec{D} = \vec{P} - \vec{Q}$, then the components are subtracted:
    • $D<em>x = P</em>x - Q_x$
    • $D<em>y = P</em>y - Q_y$
    • $D<em>z = P</em>z - Q_z$ (for 3D)
• Multiplication by a Scalar:
  • If $\vec{T} = c\vec{S}$, where $c$ is a scalar, then each component of $\vec{S}$ is multiplied by $c$:
    • $T<em>x = cS</em>x$
    • $T<em>y = cS</em>y$
    • $T<em>z = cS</em>z$ (for 3D)
• WARNING: It is critical to never attempt to add or subtract vectors as if they were scalars (i.e., simply adding or subtracting their magnitudes without considering direction). The magnitude of the sum of two vectors is generally not equal to the sum of their individual magnitudes (i.e., $|\vec{A} + \vec{B}| \neq |\vec{A}| + |\vec{B}|$ unless they are parallel and in the same direction).

Vector Equations (3.4)

• A single vector equation serves as a shorthand representation for a set of separate equations, one for each component (x, y, and z).
• Example: Newton's Second Law, $\vec{F}<em>{net} = m\vec{a}$, where the net force might be a sum of several forces like gravity, normal force, and friction ($\vec{F}</em>{net} = \vec{F}_G + \vec{n} + \vec{f}$).
• This single vector equation expands into three independent scalar equations:
  • $(\vec{F}<em>G)</em>x + (\vec{n})<em>x + (\vec{f})</em>x = m a_x$
  • $(\vec{F}<em>G)</em>y + (\vec{n})<em>y + (\vec{f})</em>y = m a_y$
  • $(\vec{F}<em>G)</em>z + (\vec{n})<em>z + (\vec{f})</em>z = m a_z$

Tilted Axes (3.4)

• For problems involving inclined planes, it is common practice to tilt the coordinate axes to simplify component resolution.
• Orientation: The x-axis is typically set parallel to the incline, and the y-axis is set normal (perpendicular) to the incline.
• Example Problem: Determine expressions for the x- and y-components of the gravitational force vector ($\vec{F}_G$) on a tilted axis where the incline angle is $\theta$.
  • If the x-axis points down the incline and the y-axis points perpendicular-outward from the incline:
    • $F<em>{Gx} = F</em>G \sin \theta$
    • $F<em>{Gy} = -F</em>G \cos \theta$
      (assuming $\vec{F}_G$ points straight down and $-y$ is into the incline).