Inner Products and Norms Calculator

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Example

Created on 2024-06-20Asked by Amelia Carter (Solvelet student)

Given vectors

\mathbf{a} = (1, 2)

and

\mathbf{b} = (3, 4)

\mathbb{R}^2

, find their inner product and the norm of

\mathbf{a}

Solution

To find the inner product of vectors

\mathbf{a} = (1, 2)

and

\mathbf{b} = (3, 4)

\mathbb{R}^2

: The inner product (dot product) of two vectors

\mathbf{a}

and

\mathbf{b}

is given by:

\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2

Substitute the components of

\mathbf{a}

and

\mathbf{b}

\mathbf{a} \cdot \mathbf{b} = 1 \cdot 3 + 2 \cdot 4

\mathbf{a} \cdot \mathbf{b} = 3 + 8

\mathbf{a} \cdot \mathbf{b} = 11

To find the norm of

\mathbf{a}

: The norm (magnitude) of a vector

\mathbf{a}

is given by:

\| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2}

Substitute the components of

\mathbf{a}

\| \mathbf{a} \| = \sqrt{1^2 + 2^2}

\| \mathbf{a} \| = \sqrt{1 + 4}

\| \mathbf{a} \| = \sqrt{5}

Therefore, the inner product of

\mathbf{a}

and

\mathbf{b}

is:

\mathbf{a} \cdot \mathbf{b} = 11

And the norm of

\mathbf{a}

is:

\| \mathbf{a} \| = \sqrt{5}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Henry Johnson on Solvelet

1. Find the norm of the vector

v = \langle 2, -3, 6 \rangle

\mathbb{R}^3

with respect to the standard inner product,2. Determine whether the integer

-28

is divisible by

4

DefinitionInner products and norms are key ideas in linear algebra and functional analysis. The most common basic operation connecting two vectors is the inner product, a more general form of the dot product, which provides a metric to compare angles and sizes in vector spaces, and has linearity, symmetry, and positive-definiteness properties. The derivative measure the length of vector, what it will give the geometrical norm, originated from the inner product. Inner products and norms allow us to examine the geometric and algebraic behaviors of spaces, such as orthogonality and distance. Example: For u,v∈Rn, ⟨u,v⟩=∑iuivi and∥u∥=⟨u,u⟩

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