02 // Visual Math & Finance SolversR⃗ = A⃗ + B⃗, A⃗ · B⃗ = |A||B| cos(Δθ), |A⃗ × B⃗| = |A||B| sin(Δθ)
Visual 2D Vector Addition Parallelogram, Dot Product & Cross Product
Given polar vectors $\vec{A} = (|A|\cos\theta_A, |A|\sin\theta_A)$ and $\vec{B} = (|B|\cos\theta_B, |B|\sin\theta_B)$, their resultant is $\vec{R} = (A_x+B_x, A_y+B_y)$ and their dot product is $\vec{A}\cdot\vec{B} = A_x B_x + A_y B_y = |A||B|\cos\Delta\theta$.
R⃗ = A⃗ + B⃗, A⃗ · B⃗ = |A||B| cos(Δθ), |A⃗ × B⃗| = |A||B| sin(Δθ)Inputs & Parameters (4)
N
1N±0.520N
°
0°±1360°
N
1N±0.520N
°
0°±1360°
Resultant |R⃗| = |A⃗ + B⃗|
16.2 @ 58.2°
Rx = 8.55, Ry = 13.76
Dot Product (A⃗ · B⃗)
18.75
Angle Δθ = 80°
2D Cross Area |A⃗ × B⃗|
106.36
Signed det = 106.36
Equilibrant Vector (−R⃗)
16.2 @ 238.2°
Balances net force to ∑F = 0
20-Row Comparison Table
Compare across:
| # | Vector |A| Magnitude (N) | Resultant Magnitude |R⃗| | Dot Product | Direction θ_R |
|---|---|---|---|---|
| 01 | 1 N | 9.226 N | 1.56 | 98.9° |
| 02 | 2 N | 9.553 N | 3.13 | 93.1° |
| 03 | 3 N | 9.969 N | 4.69 | 87.8° |
| 04 | 4 N | 10.464 N | 6.25 | 82.9° |
| 05 | 5 N | 11.029 N | 7.81 | 78.5° |
| 06 | 6 N | 11.651 N | 9.38 | 74.5° |
| 07 | 7 N | 12.324 N | 10.94 | 71° |
| 08 | 8 N | 13.039 N | 12.5 | 67.8° |
| 09 | 9 N | 13.789 N | 14.07 | 65° |
| 10 | 10 N | 14.569 N | 15.63 | 62.5° |
| 11 | 11 N | 15.375 N | 17.19 | 60.2° |
| 12 | 12 N (Current) | 16.202 N | 18.75 | 58.2° |
| 13 | 13 N | 17.048 N | 20.32 | 56.3° |
| 14 | 14 N | 17.91 N | 21.88 | 54.7° |
| 15 | 15 N | 18.785 N | 23.44 | 53.2° |
| 16 | 16 N | 19.673 N | 25.01 | 51.8° |
| 17 | 17 N | 20.57 N | 26.57 | 50.5° |
| 18 | 18 N | 21.477 N | 28.13 | 49.4° |
| 19 | 19 N | 22.392 N | 29.69 | 48.3° |
| 20 | 20 N | 23.313 N | 31.26 | 47.3° |
Was this tool helpful?
Key Analytical Takeaways
- 01The dashed parallelogram completion shows why vector addition is commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.
- 02When the angle between vectors is $90^\circ$ (orthogonal), the dot product $\vec{A}\cdot\vec{B} = 0$ and the cross-product parallelogram area reaches its maximum $|A||B|$.
- 03The scalar projection of $\vec{A}$ onto $\vec{B}$ has length $\text{comp}_{\vec{B}}\vec{A} = \frac{\vec{A}\cdot\vec{B}}{|\vec{B}|}$.
Frequently Asked Questions
What does a negative dot product tell you about two vectors?+
A negative dot product means the angle Δθ between the two vectors is obtuse (between 90° and 180°), so they point partially in opposite directions.
What is the geometric meaning of the 2D cross product |A × B|?+
The magnitude |A_x·B_y − A_y·B_x| = |A||B|sin(Δθ) equals the exact surface area of the parallelogram spanned by vectors A and B.
What does a negative dot product A⃗ · B⃗ < 0 indicate geometrically?+
Since A⃗ · B⃗ = |A||B| cos(Δθ), a negative scalar product indicates the angle between the two vectors is obtuse (between 90° and 180°), meaning they point partially in opposing directions.