Vector Inner Product Calculator Desmos

Vector Inner Product Formula:

\[ \langle u,v \rangle = u_1 \times v_1 + u_2 \times v_2 + u_3 \times v_3 \]

Vector u first component (u₁):

Vector u second component (u₂):

Vector u third component (u₃):

Vector v first component (v₁):

Vector v second component (v₂):

Vector v third component (v₃):

Inner Product Result (scalar):

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1. What is Vector Inner Product?

The inner product (or dot product) of two vectors is a scalar quantity that measures the product of their magnitudes and the cosine of the angle between them. It's fundamental in vector algebra and has wide applications in physics and engineering.

2. How Does the Calculator Work?

The calculator uses the standard inner product formula:

\[ \langle u,v \rangle = u_1 \times v_1 + u_2 \times v_2 + u_3 \times v_3 \]

Where:

\( u_1, u_2, u_3 \) — Components of vector u
\( v_1, v_2, v_3 \) — Components of vector v

Explanation: The inner product is calculated by multiplying corresponding components and summing the results.

3. Applications of Inner Product

Details: Inner products are used to calculate angles between vectors, determine orthogonality, in projections, and in many physics applications like work calculations.

4. Using the Calculator

Tips: Enter all six components (three for each vector). The calculator works with any real numbers, including decimals and negative values.

5. Frequently Asked Questions (FAQ)

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

Q2: Can this calculator handle 2D vectors?
A: Yes, simply set the third component (u₃ and v₃) to zero for 2D vectors.

Q3: What's the difference between inner product and cross product?
A: Inner product gives a scalar, while cross product gives a vector perpendicular to both input vectors.

Q4: How is this related to Desmos?
A: Desmos is a graphing calculator that can visualize vectors and compute their inner products.

Q5: What's the geometric interpretation?
A: The inner product relates to the angle θ between vectors: \( \langle u,v \rangle = \|u\| \|v\| \cos \theta \).