Vector Calculator & Visualizer
Interactive vector addition, subtraction, and dot product. Visualize magnitude and direction on a 2D grid.
Vectors are the language of physics and engineering. Use this grid to visually **Add**, **Subtract**, and find the **Dot Product** of two vectors.
Vectors
Vector Grid
Calculations
Add X components and Y components separately.
What is Vector Calculator & Visualizer?
What is a Vector?
A vector is a quantity that has both **magnitude** (length) and **direction**. It's different from a scalar (like temperature), which only has magnitude.
We represent 2D vectors as coordinates $(x, y)$, meaning "move x units right, and y units up".
Formula & Calculation
Addition (Tip-to-Tail)
Simply add matching components:
u + v = (uₓ+vₓ, uᵧ+vᵧ)Dot Product (Scalar)
Measures how much two vectors point in the same direction:
u · v = |u||v|cos(θ)Calculations
- Magnitude: $|v| = \sqrt{x^2 + y^2}$ (Pythagorean theorem)
- Direction (Angle): $\theta = \tan^ {-1} (y/x)$
Example Calculation
Physics Example: Forces
Imagine two people pushing a box.
- Person A (Blue): Pushes East with 3N force ($u = 3,0$).
- Person B (Red): Pushes North with 4N force ($v = 0,4$).
- Result: The box moves North-East with 5N force.
Use the add mode above to verify that $(3,0) + (0,4) = (3,4)$, and the magnitude is 5.
Frequently Asked Questions
What does a negative dot product mean?
It means the vectors are pointing generally in opposite directions (angle greater than 90°). If it's zero, they are perpendicular (90°).
Why x and y?
Breaking diagonal movement into horizontal (x) and vertical (y) steps makes math infinitely easier than dealing with angles and lengths directly.