Vector Multiplication Calculator

Dot Product Formula:

\[ a \cdot b = a_x b_x + a_y b_y + a_z b_z \]

Vector a (x-component):

Vector a (y-component):

Vector a (z-component):

Vector b (x-component):

Vector b (y-component):

Vector b (z-component):

Dot Product (a · b):

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1. What is Vector Dot Product?

The dot product (also known as scalar product) is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It measures the magnitude of one vector in the direction of another.

2. How Does the Calculator Work?

The calculator uses the dot product formula:

\[ a \cdot b = a_x b_x + a_y b_y + a_z b_z \]

Where:

\( a_x, a_y, a_z \) — Components of vector a
\( b_x, b_y, b_z \) — Components of vector b
\( a \cdot b \) — Resulting scalar value (unitless)

Explanation: The dot product is calculated by multiplying corresponding components of the two vectors and summing the results.

3. Importance of Dot Product Calculation

Details: The dot product is fundamental in physics, engineering, and computer graphics. It's used to calculate work done by a force, determine angles between vectors, project one vector onto another, and check for orthogonality between vectors.

4. Using the Calculator

Tips: Enter the x, y, and z components for both vectors. The calculator will compute the dot product, which is a scalar quantity with no units (unless the vector components themselves have units).

5. Frequently Asked Questions (FAQ)

Q1: What does a dot product of zero mean?
A: A dot product of zero indicates that the two vectors are perpendicular (orthogonal) to each other.

Q2: Can dot product be negative?
A: Yes, the dot product can be negative, which indicates that the angle between the vectors is greater than 90 degrees.

Q3: What's the difference between dot product and cross product?
A: Dot product gives a scalar result, while cross product gives a vector result that is perpendicular to both input vectors.

Q4: How is dot product related to vector magnitudes?
A: The dot product of a vector with itself equals the square of its magnitude: \( a \cdot a = |a|^2 \).

Q5: What are practical applications of dot product?
A: Used in physics for work calculations, in computer graphics for lighting and shading, in machine learning for similarity measures, and in engineering for force projections.