Vector Equilibrium, Matrix Systems, and Desmos Matrix Solutions

Step-by-Step System of Equations and Vector Mechanics Setup

• Initial Parameters and Vector Definitions:

  • Given tension magnitude value: $T_{AC} = 826$

  • Vector $AD$ coordinates and displacement values: $2$, $1$, $74$

  • Step-by-step vector construction process:

  • Determine the distance/dimension vector from spatial coordinates.

  • Calculate the scalar magnitude of the distance vector.

  • Divide the distance vector by its scalar magnitude to obtain the unit vector.

  • Multiply the unit vector by the scalar force variable to obtain the force vector expression.

• Equilibrium Equations Formulation:

  • Standard equilibrium condition requires the sum of all vector components along each orthogonal axis to equal zero:

  • $\sum F_i = 0$

  • $\sum F_j = 0$

  • $\sum F_k = 0$

  • System algebraic organization rule:

  • All terms containing unknown variables ($T_{AB}$, $T_{AD}$, $P$) remain on the left side of the equation.

  • All known constant numerical terms must be transposed to the right-hand side of the equal sign, reversing their algebraic signs.

  • Writing out this step explicitly on paper prevents sign errors during matrix conversion.

Summation of Vector Components

• Summation of $i$ Components ($\sum F_i = 0$):

  • Raw component equation:

  • $-T_{AB} \left(\frac{20}{105}\right) + 420 - T_{AD} \left(\frac{20}{126}\right) = 0$

  • Standard form with constant transposed to the right-hand side:

  • $-T_{AB} \left(\frac{20}{105}\right) - T_{AD} \left(\frac{20}{126}\right) + 0 \cdot P = -420$

• Summation of $j$ Components ($\sum F_j = 0$):

  • Physical origin of key values:

  • The value $100$ originates directly from the $j$ component of the distance vector measurement.

  • Raw component equation:

  • $-T_{AB} \left(\frac{100}{105}\right) - T_{AD} \left(\frac{100}{126}\right) + 1 \cdot P - 700 = 0$

  • Standard form with constant transposed to the right-hand side:

  • $-T_{AB} \left(\frac{100}{105}\right) - T_{AD} \left(\frac{100}{126}\right) + 1 \cdot P = 700$

• Summation of $k$ Components ($\sum F_k = 0$):

  • Component terms:

  • Term for $T_{AB}$: $\frac{25}{105}$

  • Term for $T_{AD}$: $-\frac{74}{126}$ (also recorded as $-\frac{74}{26}$)

  • Constant term: $127$ (transposed as $-127$)

  • Standard form with constant transposed to the right-hand side:

  • $T_{AB} \left(\frac{25}{105}\right) - T_{AD} \left(\frac{74}{126}\right) + 0 \cdot P = -127$

  • Writing out equations clearly prevents the board work from resembling a CVS receipt.

Matrix Formulation and Desmos Matrix Computation

• Augmented Matrix Structure:

  • Column structure: Matrix columns correspond sequentially to variables $T_{AB}$, $T_{AD}$, $P$, and the augmented right-hand side constant column.

  • Row 1 ($i$-component coefficients):

  • $T_{AB}$ coefficient: $-\frac{20}{105}$

  • $T_{AD}$ coefficient: $-\frac{20}{126}$

  • $P$ coefficient: $0$

  • Augmented constant value: $-420$

  • Row 2 ($j$-component coefficients):

  • $T_{AB}$ coefficient: $-\frac{100}{105}$

  • $T_{AD}$ coefficient: $-\frac{100}{126}$

  • $P$ coefficient: $1$

  • Augmented constant value: $700$

  • Row 3 ($k$-component coefficients):

  • $T_{AB}$ coefficient: $\frac{25}{105}$

  • $T_{AD}$ coefficient: $-\frac{74}{126}$

  • $P$ coefficient: $0$

  • Augmented constant value: $-127$ (entered into Desmos as $-126$)

• Matrix Solution via Desmos Matrix Tool:

  • Hardware display setup: Use rear display controls configured to the left side to view Screen 2 across back screens.

  • Procedure: Apply Reduced Row Echelon Form ($\text{RREF}$) to Matrix $A$ ($\text{RREF}(A)$).

  • Computed output vector values:

  • $T_{AB} = 1514.55\,\text{N}$

  • $T_{AD} = 828.54\,\text{N}$

  • $P = 2800\,\text{N}$

  • Physical verification:

  • Force $P$ is positive ($2800\,\text{N}$), confirming that vector $P$ points vertically upward as originally indicated on the free body diagram.

Questions and Audience Interactions

• Student seating and identity clarifications:

  • Gabriel: Identified correctly after initial confusion with Logan due to the letter L; sitting near Brandon and Vanessa.

  • Luciana: Seated directly in front of the second Brandon and Vanessa.

  • Julian: Seated adjacent to Gabriel.

• Distance vector origins and conceptual verification:

  • Question posed to virtual student Anisa: Where did the number $100$ originally originate?

  • Student contribution by Diego: Confirmed that the value $100$ originates from the distance vector measurement corresponding to the $j$ component.

• Equation manipulation dialogue:

  • Contribution by Brandon: Identified the step to transpose constant terms ($420$) across the equals sign.

  • Correction by Doctor Nicky: Confirmed the constant becomes $-420$ inside the augmented matrix.

  • Contribution by Ryan: Recorded all equations on paper throughout the lecture, earning extra credit.

• Online vs. In-Person attendance stance:

  • Nicholas Wood (Nick): Attending in-person on Wednesday, stating a personal preference: "I don't trust anything that has a transistor. I'm taking it in person."

Examination Strategy and Administrative Guidelines

• Equipment and Software Preparation:

  • Ensure computers are fully updated and Desmos Matrix or chosen calculators are downloaded prior to taking exams online or in person.

  • Complete the posted online practice module specifically to verify equipment and software settings beforehand.

• Written Work and Partial Credit Requirements:

  • Full credit on examinations requires writing out all intermediate steps, component summations, and matrix layouts.

  • Exam problems may be divided into multi-part prompts:

  • Part A: Writing out dimension/distance vectors (such as $60$, $100$, $18$) without calculating full unit vectors.

  • Part B: Formulating force vectors given tension data, summing $i$, $j$, and $k$ components, and writing out the full matrix prior to entering terms into a calculator.

• Course Administration:

  • Section 3.1 Dropbox will be opened; several assignments have already been submitted.

  • Hard copies of pre-class quizzes will be provided following printing delays.