Vector Addition: Law of Cosines and Parallelogram

Scene Overview

• Purpose: Demonstrate how the sum of two vectors can be obtained first by applying the law of cosines ("علاقة شال") and then by constructing the parallelogram of vectors ("علاقة متوازي الاضلاع"). The animation will visually confirm that both methods yield the same resultant vector.
• Duration: ~25 seconds (well within the 30‑second limit).
• Scene: A single 2‑D Cartesian plane with a fixed camera; no camera movement is required.

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1. Visual Setup (0 – 3 s)

• Axes: Light gray Cartesian axes spanning from $-5$ to $5$ on both axes.
• Vectors:
  • Vector $\mathbf{a}$: Arrow starting at the origin, length 3 units, direction $30^{\circ}$ above the positive x‑axis. Color: red.
  • Vector $\mathbf{b}$: Arrow starting at the origin, length 4 units, direction $120^{\circ}$ from the positive x‑axis (i.e., 90° above the y‑axis). Color: blue.
• Labels (optional, with opaque background): Small labels "$\mathbf{a}$" and "$\mathbf{b}$" placed near the arrowheads.

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2. Law of Cosines Method (3 – 13 s)

2.1 Form the Triangle (3 – 6 s)

• Fade in a thin gray line connecting the tip of $\mathbf{a}$ to the tip of $\mathbf{b}$, completing a triangle with the origin.
• Highlight the interior angle $\ heta$ between $\mathbf{a}$ and $\mathbf{b}$ using a curved arc (orange) and display the angle value $\ heta = 90^{\circ}$ (since the chosen directions give a right angle; adjust if you prefer a non‑right angle).

2.2 Display Law of Cosines Formula (6 – 8 s)

• Show the formula with an opaque white background:
  $$\\|\mathbf{c}\\|^{2}=\\|\mathbf{a}\\|^{2}+\\|\mathbf{b}\\|^{2}-2\\|\mathbf{a}\\|\\|\mathbf{b}\\|\cos\ heta$$
• The symbols $\\|\mathbf{a}\\|$ and $\\|\mathbf{b}\\|$ are highlighted in red and blue respectively; $\\|\mathbf{c}\\|$ will be highlighted in green.

2.3 Compute Resultant Magnitude (8 – 10 s)

• Substitute the numeric values ($\\|\mathbf{a}\\|=3$, $\\|\mathbf{b}\\|=4$, $\cos\ heta=0$ for a right angle) and animate the calculation step‑by‑step, ending with $\\|\mathbf{c}\\| = 5$.

2.4 Draw Resultant Vector (10 – 13 s)

• From the origin, draw a green arrow $\mathbf{c}$ whose tip coincides with the tip of the gray connecting line (i.e., the diagonal of the triangle). Its length is 5 units and its direction matches the geometric diagonal.
• Fade in a label "$\mathbf{c}$" near the tip.

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3. Parallelogram Method (13 – 23 s)

3.1 Construct Parallelogram (13 – 16 s)

• Duplicate $\mathbf{a}$ and translate it so its tail starts at the tip of $\mathbf{b}$ (drawn in a semi‑transparent red).
• Duplicate $\mathbf{b}$ and translate it so its tail starts at the tip of $\mathbf{a}$ (drawn in a semi‑transparent blue).
• The four sides now form a purple parallelogram.

3.2 Highlight Diagonal (16 – 18 s)

• Emphasize the diagonal from the origin to the opposite corner of the parallelogram (same as the previously drawn $\mathbf{c}$) by thickening the green arrow and adding a subtle glow.

3.3 Display Parallelogram Law (18 – 20 s)

• Show the vector equation with an opaque background:
  $$\mathbf{c}=\mathbf{a}+\mathbf{b}$$
• Optionally, also display the component form:
  $$\mathbf{c}=\begin{pmatrix}a_x+b_x\\ a_y+b_y\end{pmatrix}$$

3.4 Verify Equality of Magnitudes (20 – 23 s)

• Animate a brief overlay that compares the length of the green resultant from the law‑of‑cosines step with the diagonal of the parallelogram, confirming they are identical (both length 5).

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4. Closing Summary (23 – 25 s)

• Fade in a concise statement with opaque background:
  "Both the law of cosines and the parallelogram construction give the same resultant vector."
• Hold for a moment, then fade out all elements together, leaving a clean black screen.

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Visual & Styling Details

• Colors: Red ($\mathbf{a}$), Blue ($\mathbf{b}$), Green (resultant $\mathbf{c}$), Orange (angle arc), Purple (parallelogram fill, semi‑transparent), Gray (axes and auxiliary lines).
• Line Widths: Axes thin (1 pt), vectors medium (4 pt), auxiliary triangle line thin (2 pt), parallelogram edges medium (3 pt).
• Opacity: Duplicated vectors for the parallelogram at 40 % opacity to keep focus on the original vectors.
• Background: Solid black for contrast.
• Transitions: Use smooth fade‑in/out (0.5 s) and drawing animations (0.8 s) for arrows and lines.
• Timing: Each major block is allocated the durations indicated above, ensuring the total runtime stays under 30 seconds.

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Note: All text elements (formulas, labels, summary) are presented with an opaque white background to guarantee readability against the dark scene.