Dot Product Calculator – Guide & Formulas
Calculate the dot product (scalar product) of two vectors instantly. Free online calculator for 2D and 3D vectors with step-by-step solutions and angle between vectors.
Calculate the dot product of two vectors with our free online calculator. Enter 2D or 3D vector components to instantly get the scalar product, angle between vectors, and step-by-step solutions.
Key Takeaway
Use the free Dot Product Calculator to calculate the dot product (scalar product) of two vectors instantly. free online calculator for 2d and 3d vectors with step-by-step solutions and angle between vectors. Get instant results with step-by-step explanations.
How to Use the Dot Product Calculator
- Select vector dimension: 2D or 3D.
- Enter the components of Vector A (ax, ay, and az for 3D).
- Enter the components of Vector B (bx, by, and bz for 3D).
- Click Calculate to see the dot product, angle, and detailed steps.
The Formula
Variable Definitions
- A · B: The dot product (scalar product) of vectors A and B
- ax, ay, az: Components of vector A
- bx, by, bz: Components of vector B
- |A|: The magnitude (length) of vector A
- θ: The angle between vectors A and B
Dot Product of A = (3, 4) and B = (5, 12)
Calculate the scalar product of two 2D vectors.
- Step 1: Identify components: A = (3, 4), B = (5, 12).
- Step 2: Apply formula: A · B = (3)(5) + (4)(12) = 15 + 48 = 63.
- Step 3: Calculate magnitudes: |A| = √(9 + 16) = √25 = 5. |B| = √(25 + 144) = √169 = 13.
- Step 4: Angle: cos(θ) = 63 / (5 × 13) = 63/65 ≈ 0.9692, so θ ≈ 14.25°.
Frequently Asked Questions
What is the dot product?
The dot product (also called scalar product or inner product) of two vectors is a scalar quantity calculated by multiplying corresponding components and summing the results. It measures how much one vector extends in the direction of another.
When is the dot product zero?
The dot product is zero when the vectors are perpendicular (orthogonal) to each other. This means cos(90°) = 0, so the vectors share no common direction.
What does a positive dot product mean?
A positive dot product means the angle between vectors is less than 90°, so they point in generally the same direction. A negative dot product means the angle is greater than 90°.
Is the dot product commutative?
Yes. A · B = B · A. The order of multiplication does not matter because scalar multiplication of components is commutative.
What is the difference between dot product and cross product?
The dot product yields a scalar (single number), while the cross product yields a vector. The dot product measures projection; the cross product measures perpendicular area.
How do I find the angle between two vectors?
Use θ = arccos(A · B / (|A| × |B|)). This requires computing the dot product and both magnitudes first.
Can I compute the dot product of more than two vectors?
The dot product is defined for exactly two vectors. To combine more vectors, compute pairwise dot products or use matrix notation.
What is the geometric interpretation of the dot product?
Geometrically, A · B = |A| × |B| × cos(θ). It equals the magnitude of A times the projection of B onto A (or vice versa). It measures the "overlap" between two vectors.
Are unit vectors important for dot products?
Yes. The dot product of two unit vectors simply equals cos(θ), making it a direct measure of the angle between them. Unit vectors simplify many calculations.
How does the dot product relate to work in physics?
In physics, work = F · d = |F| × |d| × cos(θ), where F is force and d is displacement. The dot product naturally appears whenever a force acts along a direction of motion.