Formula Used
Standard real product: <a,b> = a1b1 + a2b2 + ... + anbn.
Weighted real product: <a,b> = w1a1b1 + w2a2b2 + ... + wnanbn.
Complex Hermitian product: <a,b> = conj(a1)b1 + conj(a2)b2 + ... + conj(an)bn.
Norm: ||a|| = square root of <a,a>. The angle uses the cosine ratio from the selected inner product.
How to Use This Calculator
- Enter the first vector in the Vector A box.
- Enter the second vector with the same number of values.
- Select standard, weighted, or complex mode.
- Add positive weights when weighted mode is selected.
- Choose degrees or radians for the angle output.
- Set decimal precision, then press the calculate button.
- Use CSV or PDF export after the result appears.
Example Data Table
| Mode |
Vector A |
Vector B |
Weights |
Inner Product |
Use Case |
| Standard |
2, -1, 4, 3 |
5, 0, -2, 7 |
1, 1, 1, 1 |
23 |
Basic coordinate comparison |
| Weighted |
1, 3, 2 |
4, -2, 5 |
2, 1, 3 |
32 |
Scaled coordinate importance |
| Complex |
1+2i, 3-i |
2-i, 4+5i |
Not used |
7 + 14i |
Hermitian vector spaces |
Understanding Inner Products
An inner product measures how two vectors relate. It extends ordinary multiplication into vector spaces. For real vectors, it is often the dot product. For complex vectors, it uses a conjugate term. The value can show alignment, length, angle, and projection. A positive value means both vectors point in a similar direction. A negative value means they lean apart. A zero value often means orthogonality. That means the vectors meet at a right angle.
Why This Calculator Helps
Manual vector work becomes slow as dimensions grow. A small sign error can change the final result. This calculator separates every term. It checks dimensions before calculation. It also supports weighted products. Weighted products are useful when some coordinates matter more than others. Complex mode is useful in linear algebra, signals, physics, and engineering. The output includes norms, cosine value, angle, projection coefficient, and downloadable reports.
Using Weighted Inner Products
A weighted inner product multiplies each coordinate pair by a weight. The weights should usually be positive. Positive weights keep the length meaning valid. If a weight is larger, that coordinate has more influence. This is helpful in data scoring and scaled spaces. It can also model custom geometry. Always use one weight for each vector position. Missing weights are not guessed by the calculator.
Reading The Results
The inner product is the main answer. Norms show each vector length under the selected rule. The cosine value compares direction. The angle converts that comparison into degrees or radians. The projection coefficient shows how much of the second vector lies along the first vector. When the product is near zero, the vectors are nearly orthogonal. Use the precision box to control rounded output. Exports help save the work for assignments or records.
Best Practice
Keep vector entries consistent. Use commas, spaces, or lines between values. Do not mix real and complex modes unless needed. In complex mode, write values like 3+2i, -4i, or 5. Review the example table before entering long vectors. For formal work, copy the formula and steps from the result panel. This gives a clear trace for later checking. It also helps explain the calculation to another reader. Careful entry makes each exported result easier to trust.
FAQs
What is an inner product?
An inner product is a rule that multiplies two vectors and returns one value. It helps measure length, angle, projection, and orthogonality in vector spaces.
Is the dot product an inner product?
Yes. The ordinary dot product is the most common real inner product. It multiplies matching coordinates and adds those products together.
When should I use weighted mode?
Use weighted mode when each coordinate has different importance. Each weight scales one coordinate pair. Positive weights keep the result valid as an inner product.
Can I enter complex vectors?
Yes. Choose complex Hermitian mode. Enter values like 3+2i, -4i, 6, or 1-i. Do not place spaces inside one complex number.
What does a zero inner product mean?
A zero inner product usually means the vectors are orthogonal under the selected rule. In real space, this often means a right angle.
Why must vector dimensions match?
Each coordinate in the first vector must pair with one coordinate in the second vector. Unequal dimensions leave unmatched values and break the formula.
What is the projection coefficient?
It tells how much of vector B lies along vector A. Multiplying vector A by this coefficient gives the projection vector.
What is included in the downloads?
The CSV and PDF files include the selected mode, dimension, inner product, norms, angle, projection, orthogonal check, and formula summary.