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Inner Product

Inner Product
Scalar Product

The general name for a vector product resulting in a scalar. The dot product is a simple example.

 

See also

Cross product

Key Formula

⟨u,v⟩=u1v1+u2v2+⋯+unvn\langle \mathbf{u}, \mathbf{v} \rangle = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n
Where:
  • u,v\mathbf{u}, \mathbf{v} = Two vectors in \mathbb{R}^n
  • ui,viu_i, v_i = The ith components of vectors u and v
  • nn = The dimension of the vector space

Worked Example

Problem: Find the inner product of u = (3, −1, 4) and v = (2, 5, −2) in ℝ³.
Step 1: Multiply corresponding components of the two vectors.
3×2=6,(−1)×5=−5,4×(−2)=−83 \times 2 = 6, \quad (-1) \times 5 = -5, \quad 4 \times (-2) = -8
Step 2: Add the products together to get the scalar result.
⟨u,v⟩=6+(−5)+(−8)=−7\langle \mathbf{u}, \mathbf{v} \rangle = 6 + (-5) + (-8) = -7
Answer: The inner product ⟨u, v⟩ = −7.

Another Example

Problem: Determine whether the vectors u = (1, 2) and v = (−4, 2) are orthogonal using the inner product.
Step 1: Compute the inner product of u and v.
⟨u,v⟩=(1)(−4)+(2)(2)=−4+4=0\langle \mathbf{u}, \mathbf{v} \rangle = (1)(-4) + (2)(2) = -4 + 4 = 0
Step 2: Two vectors are orthogonal (perpendicular) if and only if their inner product equals zero. Since the result is 0, the vectors are orthogonal.
Answer: Yes, u and v are orthogonal because ⟨u, v⟩ = 0.

Frequently Asked Questions

What is the difference between an inner product and a dot product?
The dot product is one specific inner product — the standard one used in ℝⁿ where you multiply corresponding components and sum. An inner product is the broader concept: any operation on a vector space that satisfies certain axioms (linearity, symmetry, and positive-definiteness). So every dot product is an inner product, but not every inner product is the dot product.
What does the inner product tell you geometrically?
The inner product measures how much two vectors point in the same direction. When the inner product is positive, the vectors point roughly the same way; when it is negative, they point roughly opposite. When the inner product is zero, the vectors are perpendicular (orthogonal). In ℝⁿ with the standard inner product, ⟨u, v⟩ = ‖u‖ ‖v‖ cos θ, where θ is the angle between them.

Inner Product (Dot Product) vs. Cross Product

The inner product takes two vectors and returns a scalar that captures how aligned the vectors are. The cross product takes two vectors in ℝ³ and returns a new vector that is perpendicular to both inputs, with magnitude representing the area of the parallelogram they span. The inner product is defined in any dimension; the cross product is specific to three dimensions (and, in a generalized form, seven dimensions).

Why It Matters

The inner product is foundational across mathematics, physics, and engineering. It provides the mechanism for defining lengths (norms), angles, and orthogonality in vector spaces. Applications range from projecting one vector onto another in physics to computing similarity between data vectors in machine learning and statistics.

Common Mistakes

Mistake: Thinking the inner product of two vectors gives another vector.
Correction: The inner product always produces a scalar (a single number), not a vector. If you got a vector, you may have computed the cross product instead.
Mistake: Forgetting to account for negative components when multiplying.
Correction: Pay careful attention to signs in each component product. A sign error in any term will give the wrong scalar and can flip whether vectors appear orthogonal or not.

Related Terms

  • Dot Product — The standard inner product in ℝⁿ
  • Cross Product — Vector operation producing a perpendicular vector
  • Scalar — The type of value an inner product returns
  • Vector — The inputs to an inner product
  • Orthogonal — Vectors with inner product equal to zero
  • Norm — Vector length defined via inner product
  • Projection — Computed using inner products