Vector Calculator

Vector Operations:

\[ \begin{align*} \text{Dot Product} & : \mathbf{v} \cdot \mathbf{w} = \sum v_i w_i \\ \text{Cross Product} & : \mathbf{v} \times \mathbf{w} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{vmatrix} \\ \text{Addition} & : \mathbf{v} + \mathbf{w} = (v_1 + w_1, v_2 + w_2, v_3 + w_3) \\ \text{Magnitude} & : \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} \end{align*} \]

Result:

Unit Converter ▲

Unit Converter ▼

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1. What are Vector Operations?

Vector operations are mathematical operations performed on vectors, which are quantities having both magnitude and direction. Common operations include addition, dot product, cross product, and magnitude calculation.

2. How Does the Calculator Work?

The calculator performs various vector operations:

Where:

\( \mathbf{v}, \mathbf{w} \) — Input vectors
\( v_i, w_i \) — Vector components
\( \mathbf{i}, \mathbf{j}, \mathbf{k} \) — Unit vectors in 3D space

3. Importance of Vector Calculations

Details: Vector operations are fundamental in physics, engineering, computer graphics, and machine learning. They describe forces, velocities, and other directional quantities.

4. Using the Calculator

Tips: Enter vectors as comma-separated values (e.g., "1,2,3"). For cross product, use 3D vectors. All components should be numeric values.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot and cross product?
A: Dot product gives a scalar (magnitude only), while cross product gives a vector perpendicular to both input vectors (3D only).

Q2: Can I calculate magnitude for any dimension?
A: Yes, magnitude works for vectors of any dimension (2D, 3D, etc.).

Q3: Why is cross product limited to 3D?
A: The cross product is only defined for three-dimensional space in standard vector algebra.

Q4: How is vector addition performed?
A: Vectors are added component-wise (x with x, y with y, etc.).

Q5: What does the dot product represent?
A: The dot product measures how much two vectors point in the same direction, and equals the product of their magnitudes times the cosine of the angle between them.