Multiplying Vectors Calculator

Vector Multiplication:

\[ \text{Dot Product: } u \cdot v = u_x v_x + u_y v_y + u_z v_z \] \[ \text{Cross Product: } u \times v = \begin{vmatrix} i & j & k \\ u_x & u_y & u_z \\ v_x & v_y & v_z \end{vmatrix} \]

Operation:

Vector u (x,y,z):

Vector v (x,y,z):

Result:

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1. What is Vector Multiplication?

Vector multiplication can produce either a scalar (dot product) or another vector (cross product). The dot product measures the similarity between vectors, while the cross product produces a vector perpendicular to both input vectors.

2. How Does the Calculator Work?

The calculator performs two types of vector multiplication:

\[ \text{Dot Product: } u \cdot v = u_x v_x + u_y v_y + u_z v_z \] \[ \text{Cross Product: } u \times v = \begin{vmatrix} i & j & k \\ u_x & u_y & u_z \\ v_x & v_y & v_z \end{vmatrix} \]

Where:

\( u \) — First vector with components (uₓ, uᵧ, u_z)
\( v \) — Second vector with components (vₓ, vᵧ, v_z)
Dot product — Results in a scalar value
Cross product — Results in a new vector

3. Dot Product vs Cross Product

Dot Product: Measures how much two vectors point in the same direction. Used in projections and calculating angles between vectors.

Cross Product: Produces a vector perpendicular to both input vectors. Used in torque calculations and finding normal vectors.

4. Using the Calculator

Steps: Select operation type (dot or cross), enter components for both vectors, and click Calculate. Results will show either a scalar value or vector components.

5. Frequently Asked Questions (FAQ)

Q1: When should I use dot product vs cross product?
A: Use dot product for projection/angle calculations. Use cross product when you need a perpendicular vector or calculating torque.

Q2: What does a dot product of zero mean?
A: Vectors are perpendicular (90° angle between them).

Q3: What does a cross product of zero mean?
A: Vectors are parallel (0° angle between them).

Q4: Can I use 2D vectors with this calculator?
A: Yes, just set the z-components to 0.

Q5: How is the cross product direction determined?
A: By the right-hand rule - point fingers along first vector, curl toward second vector, thumb points in cross product direction.