In the $\mathbb{R}^4$ example, how are the norms of $u$ and $v$ calculated?

For $u=(1,0,0,1)$, $\|u\|=\sqrt{1^2+0^2+0^2+1^2}=\sqrt{2}$. For $v=(0,2,0,0)$, $\|v\|=\sqrt{0^2+2^2+0^2+0^2}=2$. The larger norm indicates $v$ is longer than $u$.

Linear Algebra 9 | Inner Product and Norm [dark version]

Q: How does an inner product determine when two vectors are orthogonal?

Orthogonality is encoded by the inner product value: if $\langle u,v\rangle=0$, then $u$ and $v$ are orthogonal (perpendicular). In $\mathbb{R}^4$, the example $u=(1,0,0,1)$ and $v=(0,2,0,0)$ has $\langle u,v\rangle=1\cdot 0+0\cdot 2+0\cdot 0+1\cdot 0=0$, so the vectors are orthogonal.

Q: What are the three defining properties of an inner product on $\mathbb{R}^n$?

The inner product must be (1) positive definite: $\langle u,u\rangle\ge 0$ and $\langle u,u\rangle=0$ iff $u$ is the zero vector; (2) symmetric: $\langle u,v\rangle=\langle v,u\rangle$; and (3) linear in the second argument: $\langle u,v+w\rangle=\langle u,v\rangle+\langle u,w\rangle$ and $\langle u,\lambda v\rangle=\lambda\langle u,v\rangle$. Symmetry then yields linearity in the first argument as well.

Q: How is the standard inner product on $\mathbb{R}^n$ computed from vector components?

If $u=(u_1,\dots,u_n)$ and $v=(v_1,\dots,v_n)$, then $\langle u,v\rangle=\sum_{i=1}^n u_i v_i$. It multiplies corresponding components and adds the results, extending the familiar dot-product formulas from $\mathbb{R}^2$ and $\mathbb{R}^3$.

Q: How does the norm relate to the inner product, and why does it produce a length?

The norm is defined by $\|u\|=\sqrt{\langle u,u\rangle}$. Since positive definiteness guarantees $\langle u,u\rangle\ge 0$, the square root is real and nonnegative. For the standard inner product, $\langle u,u\rangle$ becomes a sum of squared components, so $\|u\|$ matches Euclidean length.

TL;DR

An inner product $⟨ u, v ⟩$ turns vector spaces like $R^{n}$ into spaces where angles and perpendicularity can be expressed algebraically.

Briefing Cornell Notes

Briefing

Inner products and norms add the missing “geometry layer” to vector spaces like $R^{n}$ : they turn raw addition and scaling into tools for measuring angles, orthogonality, and lengths. With an inner product, two vectors $u$ and $v$ produce a real number $⟨ u, v ⟩$ that encodes whether they are perpendicular—specifically, $⟨ u, v ⟩ = 0$ means the vectors are orthogonal. This is the same geometric idea familiar from $R^{2}$ and $R^{3}$ , but it extends cleanly to higher dimensions, where visual intuition fades and algebraic definitions matter more.

The standard inner product on $R^{n}$ is defined component-by-component: if $u = (u_{1}, \dots, u_{n})$ and $v = (v_{1}, \dots, v_{n})$ , then $⟨ u, v ⟩ = i = 1 \sum n u_{i} v_{i} .$ This construction generalizes the familiar $u_{1} v_{1} + u_{2} v_{2}$ from $R^{2}$ and $u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3}$ from $R^{3}$ . Beyond the formula, the inner product is characterized by three key properties. First, it is positive definite: $⟨ u, u ⟩ \geq 0$ for every vector, and $⟨ u, u ⟩ = 0$ happens only when $u$ is the zero vector—because $⟨ u, u ⟩$ becomes a sum of squares. Second, it is symmetric: $⟨ u, v ⟩ = ⟨ v, u ⟩$ , reflecting commutativity of real multiplication. Third, it is linear in the second argument: $⟨ u, v + w ⟩ = ⟨ u, v ⟩ + ⟨ u, w ⟩$ and $⟨ u, λ v ⟩ = λ ⟨ u, v ⟩$ for scalars $λ$ . Symmetry then implies the same linear behavior in the first argument as well.

Once an inner product exists, a norm follows naturally as a length measure. The standard norm of $u$ is defined by $∥ u ∥ = ⟨ u, u ⟩ .$ Because $⟨ u, u ⟩$ is nonnegative, the square root produces a real, nonnegative length. In $R^{2}$ and $R^{3}$ , this reproduces the usual Euclidean distance formula—essentially the Pythagorean theorem in algebraic form—while in higher dimensions it becomes the same “sum of squared components, then square root” rule.

A concrete example in $R^{4}$ makes the geometry tangible. For $u = (1, 0, 0, 1)$ and $v = (0, 2, 0, 0)$ , the inner product is zero, so the vectors are orthogonal (a right angle in spirit, even if not drawn). Their lengths come from the norm: $∥ u ∥ = 1^{2} + 1^{2} = 2$ and $∥ v ∥ = 2^{2} = 2$ , showing that $v$ is longer. The takeaway is that these definitions preserve familiar geometric meaning while scaling to $R^{n}$ , setting up later discussion of more abstract (and even “strange”) geometries built from the same underlying inner-product properties.

Cornell Notes

An inner product on $R^{n}$ assigns a real number $⟨ u, v ⟩$ to two vectors, letting orthogonality and angle-related geometry be expressed algebraically. The standard inner product is $⟨ u, v ⟩ = \sum_{i = 1}^{n} u_{i} v_{i}$ , and it satisfies positive definiteness ( $⟨ u, u ⟩ \geq 0$ , with equality only for the zero vector), symmetry ( $⟨ u, v ⟩ = ⟨ v, u ⟩$ ), and linearity in the second argument. A norm then measures length via $∥ u ∥ = ⟨ u, u ⟩$ , generalizing Euclidean length and the Pythagorean theorem. This framework makes it possible to talk about angles, perpendicular vectors, and lengths even in high-dimensional spaces like $R^{4}$ .

How does an inner product determine when two vectors are orthogonal?

Orthogonality is encoded by the inner product value: if

⟨ u, v ⟩ = 0

, then

u

and

v

are orthogonal (perpendicular). In

R^{4}

, the example

u = (1, 0, 0, 1)

and

v = (0, 2, 0, 0)

has

⟨ u, v ⟩ = 1 \cdot 0 + 0 \cdot 2 + 0 \cdot 0 + 1 \cdot 0 = 0

, so the vectors are orthogonal.

What are the three defining properties of an inner product on $R^{n}$ ?

The inner product must be (1) positive definite:

⟨ u, u ⟩ \geq 0

and

⟨ u, u ⟩ = 0

iff

u

is the zero vector; (2) symmetric:

⟨ u, v ⟩ = ⟨ v, u ⟩

; and (3) linear in the second argument:

⟨ u, v + w ⟩ = ⟨ u, v ⟩ + ⟨ u, w ⟩

and

⟨ u, λ v ⟩ = λ ⟨ u, v ⟩

. Symmetry then yields linearity in the first argument as well.

How is the standard inner product on $R^{n}$ computed from vector components?

u = (u_{1}, \dots, u_{n})

and

v = (v_{1}, \dots, v_{n})

, then

⟨ u, v ⟩ = \sum_{i = 1}^{n} u_{i} v_{i}

. It multiplies corresponding components and adds the results, extending the familiar dot-product formulas from

R^{2}

and

R^{3}

How does the norm relate to the inner product, and why does it produce a length?

The norm is defined by

∥ u ∥ = ⟨ u, u ⟩

. Since positive definiteness guarantees

⟨ u, u ⟩ \geq 0

, the square root is real and nonnegative. For the standard inner product,

⟨ u, u ⟩

becomes a sum of squared components, so

∥ u ∥

matches Euclidean length.

In the $R^{4}$ example, how are the norms of $u$ and $v$ calculated?

For

u = (1, 0, 0, 1)

∥ u ∥ = 1^{2} + 0^{2} + 0^{2} + 1^{2} = 2

. For

v = (0, 2, 0, 0)

∥ v ∥ = 0^{2} + 2^{2} + 0^{2} + 0^{2} = 2

. The larger norm indicates

v

is longer than

u

Review Questions

Given vectors $u$ and $v$ in $R^{n}$ , what condition on $⟨ u, v ⟩$ guarantees they are orthogonal?
Write the formula for the standard inner product on $R^{n}$ and identify which property ensures $⟨ u, u ⟩$ cannot be negative.
How is the norm $∥ u ∥$ defined in terms of the inner product, and what familiar geometric rule does it generalize?

Key Points

1
An inner product $⟨ u, v ⟩$ turns vector spaces like $R^{n}$ into spaces where angles and perpendicularity can be expressed algebraically.
2
The standard inner product on $R^{n}$ is $⟨ u, v ⟩ = \sum_{i = 1}^{n} u_{i} v_{i}$ .
3
Orthogonality is characterized by $⟨ u, v ⟩ = 0$ .
4
A valid inner product must be positive definite, symmetric, and linear in the second argument (with symmetry implying linearity in the first).
5
The standard norm is defined by $∥ u ∥ = ⟨ u, u ⟩$ , producing a nonnegative length.
6
In $R^{4}$ , orthogonality and lengths can be computed directly from components, even when geometry is hard to visualize.
7
These definitions set up later generalizations to more abstract geometries beyond the standard Euclidean one.

Highlights

Orthogonality becomes a single algebraic test:

⟨ u, v ⟩ = 0

The standard inner product in Rn is just a component-wise multiply-and-sum: ∑i=1n​ui​vi​.

Lengths come from the inner product via ∥u∥=⟨u,u⟩​, turning sums of squares into Euclidean distance.

In R4, (1,0,0,1) and (0,2,0,0) are orthogonal, with norms 2​ and 2 respectively.

Topics

Inner Product
Norm
Orthogonality
Euclidean Geometry
Vector Spaces