Vector Operations, Component Analysis, and System of Vector Equations

Course Deadlines and Assignment Schedule

• Initial Homework Assignments (First Four Assignments):
  • Computer scores and stress dot dot files are typically due by 8:00 AM on the Monday following their assigned week.
  • Due to the Labor Day holiday on Monday, an extra day extension is granted.
  • All of the first four assignments are officially due on Tuesday after Labor Day at 8:00 AM.
• Subsequent Homework Assignments:
  • Math Review (Quadratic Equations): Focuses on quadratic equations as applied to kinematics problem-solving.
  • Vector Addition Assignment: Covers addition of vector sets.
  • Both assignments are officially due on Friday, September 11th, with computer score files due by Monday, September 14th at 8:00 AM.
• Supplemental Problem 1b:
  • Covers simultaneous vector equations, vector addition, and vector subtraction.
  • Officially due on Wednesday, September 9th.

Graphical and Analytical Vector Addition

• Graphical Rules for Adding Vectors:
  • Vectors must be drawn sequentially in a tail-to-tip orientation.
  • The tip of the preceding vector serves as the origin for the next vector in the sequence.
  • The resultant vector mathbf{R} represents the total vector sum, drawn from the tail of the first vector directly to the tip of the final vector.
• Analytical Component Method:
  • Vectors given in polar form cannot be summed directly by their magnitudes and angles; they must first be converted to rectangular ($x$ and $y$) components.
  • $x$-component calculation: R_x = |mathbf{R}| \times cos(theta)
  • $y$-component calculation: R_y = |mathbf{R}| \times sin(theta)
• Detailed Addition Example (Dog Walk Scenario):
  • First displacement vector mathbf{r}_1: Magnitude of 20,ft at an angle of 30^circ.
    • r_{1x} = 20,ft \times cos(30^circ) = 17.32,ft
    • r_{1y} = 20,ft \times sin(30^circ) = 10.00,ft (noting that sin(frac{pi}{6}) = 0.5)
  • Second displacement vector mathbf{r}_2: Magnitude of 15,ft at an angle of 115^circ relative to origin.
    • r_{2x} = 15,ft \times cos(115^circ) = -6.34,ft
    • r_{2y} = 15,ft \times sin(115^circ) = 13.60,ft
  • Summing rectangular components:
    • R_x = r_{1x} + r_{2x} = 17.32,ft + (-6.34,ft) = 10.98,ft
    • R_y = r_{1y} + r_{2y} = 10.00,ft + 13.60,ft = 23.60,ft
  • Reconstructing the Resultant Vector mathbf{R} in Polar Form:
    • Magnitude: |mathbf{R}| = sqrt{R_x^2 + R_y^2} = sqrt{(10.98,ft)^2 + (23.60,ft)^2} = 26.03,ft
    • Direction Angle: theta = tan^{-1}left(frac{R_y}{R_x}right) = tan^{-1}left(frac{23.60,ft}{10.98,ft}right) = 65.0^circ
  • Radical Notation Convention: A vertical line added at the termination of a radical symbol can be used to explicitly delineate where the expression under the radical ends.

Vector Subtraction and Displacement

• Physical Definition of Displacement:
  • Displacement is defined as the net change in position: Delta mathbf{r}.
  • By standard physical definition, any change in a quantity denoted by Delta is calculated as final value minus initial value: Delta mathbf{r} = mathbf{r}_f - mathbf{r}_i
• Vector Subtraction Framework:
  • Vector subtraction is identical to vector addition, using the additive inverse of the initial vector: mathbf{r}_f - mathbf{r}_i = mathbf{r}_f + (-mathbf{r}_i).
  • Multiplying a vector by $-1$ preserves its magnitude while flipping its direction by 180^circ.
  • Graphically, the displacement vector Delta mathbf{r} points directly from the position tip of the initial vector mathbf{r}_i to the position tip of the final vector mathbf{r}_f
• Detailed Subtraction Example (Displacement Calculation):
  • Initial position vector: mathbf{r}_i = 20,ft at 30^circ (r_{ix} = 17.32,ft, r_{iy} = 10.00,ft)
  • Final position vector: mathbf{r}_f = 26.03,ft at 65.0^circ (r_{fx} = 10.98,ft approx 11,ft, r_{fy} = 23.60,ft)
  • Calculating components of displacement Delta mathbf{r} = mathbf{r}_f - mathbf{r}_i:
    • Delta r_x = r_{fx} - r_{ix} = 10.98,ft - 17.32,ft = -6.34,ft
    • Delta r_y = r_{fy} - r_{iy} = 23.60,ft - 10.00,ft = 13.60,ft
  • Reconstructing Displacement Magnitude and Angle:
    • Magnitude: |Delta mathbf{r}| = sqrt{(-6.34,ft)^2 + (13.60,ft)^2} = 15.00,ft
    • Raw inverse tangent calculation: theta_{raw} = tan^{-1}left(frac{13.60,ft}{-6.34,ft}right) = -65.0^circ
    • Polar Quadrant Correction Rule: Always check the sign of the $x$-component when converting to polar coordinates. Because the $x$-component is negative (-6.34,ft), 180^circ must be added to the output angle: theta = -65.0^circ + 180^circ = 115^circ.

Multiplication of a Vector by a Scalar

• Concept of Scalar Multiplication:
  • Scalar multiplication entails scaling a vector by a pure number or a scalar physical quantity carrying units (e.g., mass in kilograms multiplied by acceleration).
  • Multiplication acts as repeated vector addition: 3mathbf{r} = mathbf{r} + mathbf{r} + mathbf{r}.
  • For a vector mathbf{r} = 3,m at 28^circ, the product 3mathbf{r} produces 9,m at 28^circ.
• Rules for Polar Form:
  • Magnitude Rule: Multiply the magnitude of the vector by the absolute value of the scalar $c$: |mathbf{R}_{new}| = |c| \times |mathbf{R}_{old}|. Vector magnitudes are strictly non-negative.
  • Direction Angle Rule:
    • If scalar $c > 0$: Retain the original direction angle.
    • If scalar $c < 0$: Add 180^circ to the original direction angle.
• Rules for Component / Rectangular Form:
  • Multiply each component independently by scalar $c$:
    • $R_{new, x} = c \times R_{old, x}$
    • $R_{new, y} = c \times R_{old, y}$
  • The signs of components automatically update directional orientation without requiring manual angle additions.

Vector Multiplication Methods and Unit Vector Notation

• Two Primary Types of Vector Multiplication:
  • Dot Product (Scalar Product): Multiplies two vectors to produce a scalar quantity.
  • Cross Product (Vector Product): Multiplies two vectors to produce a new vector quantity perpendicular to the plane of the original vectors.
• Unit Vector Notation Fundamentals:
  • Unit vectors are defined as dimensionless vectors of magnitude equal to exactly one (|hat{i}| = 1, |hat{j}| = 1).
  • Represented using a caret symbol (hat{}) above the variable.
  • hat{i} points exclusively along the positive $x$-axis.
  • hat{j} points exclusively along the positive $y$-axis.
  • hat{k} points exclusively along the positive $z$-axis in three-dimensional systems.
• Expressing Vectors via Unit Vectors:
  • A vector with rectangular components $R_x$ and $R_y$ is written as mathbf{R} = R_xhat{i} + R_yhat{j}.
  • Example: A position vector 2,m,hat{i} - 6,m,hat{j} specifies 2,m along the positive $x$-axis and 6,m along the negative $y$-axis.

Supplemental Problem 1b: System of Vector Equations

• Problem Statement and Given Parameters:
  • Two unknown position vectors: mathbf{r}_1 and mathbf{r}_2.
  • Condition 1: Walking along mathbf{r}_1 and then along mathbf{r}_2 leads to resultant vector mathbf{a} = 20.6,m at 14^circ.
    • mathbf{r}_1 + mathbf{r}_2 = mathbf{a}
  • Condition 2: Walking along three times mathbf{r}_1 and then along twice the magnitude of mathbf{r}_2 in the opposite direction leads to resultant vector mathbf{b} = 40,m at 90^circ.
    • 3mathbf{r}_1 - 2mathbf{r}_2 = mathbf{b}
• Algebraic Elimination Procedure for Vector Variables:
  • Step 1: Multiply the first vector equation by 2:
    • 2mathbf{r}_1 + 2mathbf{r}_2 = 2mathbf{a}
  • Step 2: Add this equation to the second vector equation to eliminate mathbf{r}_2:
    • (2mathbf{r}_1 + 2mathbf{r}_2) + (3mathbf{r}_1 - 2mathbf{r}_2) = 2mathbf{a} + mathbf{b}
    • 5mathbf{r}_1 = 2mathbf{a} + mathbf{b}
  • Step 3: Isolate vector mathbf{r}_1:
    • mathbf{r}_1 = frac{1}{5}(2mathbf{a} + mathbf{b}) = frac{2}{5}mathbf{a} + frac{1}{5}mathbf{b}
  • Step 4: Resolve vectors mathbf{a} and mathbf{b} into components to compute values for mathbf{r}_1 and subsequently isolate mathbf{r}_2 = mathbf{a} - mathbf{r}_1, finally converting both back into polar coordinates.

Standard Seven-Step Problem-Solving Protocol

• Step 1: State or transcribe the problem clearly at the top of the assignment.
• Step 2: Explicitly record all Given Information (numerical values, directions, and units).
• Step 3: Clearly define What You Are Looking For (target variable names and descriptions).
• Step 4: Provide a Diagram to visually clarify the problem setup for the reader.
• Step 5: State the Attack Strategy (written explanation of principles, equations, and algebra steps to be applied).
• Step 6: Execute and display all Detailed Work cleanly and step-by-step.
• Step 7: Perform an Answer Check (verify units, numerical reasonableness, limiting cases, and vector quadrants).