How To Multiply Vectors Calculator

Vector Multiplication:

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

Operation:

Vector u:

Vector v:

Result:

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1. What is Vector Multiplication?

Vector multiplication refers to two distinct operations that can be performed on vectors: the dot product (scalar product) and the cross product (vector product). These operations have different geometric interpretations and applications in physics and engineering.

2. Types of Vector Multiplication

There are two main types of vector multiplication:

Dot Product (Scalar Product): Results in a scalar quantity. Measures the projection of one vector onto another.
Cross Product (Vector Product): Results in a vector quantity. Produces a vector perpendicular to both input vectors.

3. Dot Product Calculation

\[ \mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y + u_z v_z \]

The dot product can also be calculated as: \[ \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta \] where θ is the angle between the vectors.

4. Cross Product Calculation

\[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_x & u_y & u_z \\ v_x & v_y & v_z \end{vmatrix} \]

The cross product results in a vector that is perpendicular to both u and v, with magnitude equal to the area of the parallelogram formed by u and v.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot and cross product?
A: Dot product gives a scalar (number) while cross product gives a vector. Dot product measures parallelism, cross product measures perpendicularity.

Q2: When is the dot product zero?
A: When vectors are perpendicular (θ = 90°), as cos(90°) = 0.

Q3: What does a zero cross product mean?
A: It means the vectors are parallel (θ = 0° or 180°), as sin(0°) = sin(180°) = 0.

Q4: Can I do cross product in 2D?
A: Yes, by treating them as 3D vectors with z=0. The result will be along the z-axis.

Q5: What are practical applications?
A: Dot product: work calculation, projections. Cross product: torque, finding normals, rotational effects.