Use our free Dot Product Calculator to find the scalar (inner) product of two or more vectors with steps. Enter vectors in any format [x, y, z] or (x, y, z) & get instant results.
Dot Product Calculator
Example:
Vector x: 3, 5, 8
Vector y: 2, 7, 1
Instantly Compute and Understand Vector Dot Products
Are you working with vectors in mathematics, physics, or engineering? Whether you’re a student, educator, or professional, our Dot Product Calculator is the perfect tool to quickly compute the dot product of two vectors and understand the process step by step.
What is the Dot Product?
The dot product (also called the scalar product or inner product) is a fundamental operation in vector mathematics. It takes two equal-length vectors and returns a single number (a scalar). This operation is widely used in geometry, physics, engineering, and computer graphics.
Dot Product Formula (Cartesian Coordinates):
For vectors
u = ⟨u₁, u₂, u₃⟩
v = ⟨v₁, v₂, v₃⟩
The dot product is:
u · v = u₁v₁ + u₂v₂ + u₃v₃
Dot Product Formula (Using Magnitude and Angle):
If you know the magnitudes of the vectors and the angle θ between them:
u · v = |u| × |v| × cos(θ)
How to Use the Dot Product Calculator
Step 1: Enter Your Vectors
- In the input fields, enter the components of each vector, separated by commas.
- Example:
- Vector x: 3, 5, 8
- Vector y: 2, 7, 1
- You can also enter vectors in the form ai + bj + ck by converting to coordinate form.
Step 2: Select Significant Figures (Optional)
- Choose how many significant figures you want your result to be rounded to, or leave it as “Auto” for full precision.
Step 3: Calculate
- Click the Calculate Dot Product button.
- Instantly, you’ll see:
- Your input vectors in mathematical notation
- The step-by-step calculation process
- The final answer
Step 4: Clear and Repeat
- To perform another calculation, click the Clear button and enter new vectors.
Example Calculation
Input:
- Vector x: ⟨3, 5, 8⟩
- Vector y: ⟨2, 7, 1⟩
Step-by-Step Solution:
The dot product is given by
u · v = Σ uᵢvᵢ
So,
⟨3, 5, 8⟩ · ⟨2, 7, 1⟩ = (3×2) + (5×7) + (8×1)
= 6 + 35 + 8
= 49
Geometric Interpretation: What Does the Dot Product Mean?
The dot product gives you a sense of how much two vectors point in the same direction.
- If the dot product is positive, the vectors point in a similar direction (angle < 90°).
- If it’s zero, the vectors are perpendicular (orthogonal).
- If it’s negative, the vectors point in opposite directions (angle > 90°).
Graphically, the dot product equals the length of the projection of one vector onto the other, multiplied by the magnitude of the other vector.
Mathematically:
u · v = |u| × |v| × cos(θ)
So, you can also use the dot product to find the angle between two vectors:
cos(θ) = (u · v) / (|u| × |v|)
Why You Use Our Dot Product Calculator?
- Fast & Accurate: Get instant, error-free results.
- Step-by-Step Solution: See exactly how the calculation is done.
- User-Friendly: Simple interface, mobile-friendly, and easy to use.
- Versatile: Works with 2D, 3D, or higher-dimensional vectors.
- Educational: Great for learning, homework, or teaching.
Frequently Asked Questions
What’s the difference between dot product and cross product?
The dot product returns a scalar (number) and measures how much two vectors point in the same direction. The cross product returns a vector that is perpendicular to both original vectors.
Can I enter vectors in i, j, k format?
Yes! Just convert them to coordinate form before entering (e.g., 2i + 3j + 4k becomes 2, 3, 4).
Does this calculator work for 2D and 3D vectors?
Absolutely! Just enter the correct number of components for each vector.
How is the dot product calculated?
By multiplying corresponding components and summing the results:
u · v = u₁v₁ + u₂v₂ + u₃v₃
Try the Dot Product Calculator Now!
Enter your vectors above and see the result instantly, complete with a clear, step-by-step solution. Whether you’re checking homework, teaching, or working on a project, this tool will save you time and help you understand vector operations.
Start calculating dot products with confidence and ease!
If you have questions or feedback about this tool, let us know in the comments below!