dot product

Meanings

• noun

  The dot product of two vectors is a scalar value found by multiplying the corresponding entries of the vectors and summing the results.

  - "The dot product of vectors u and v is denoted as u · v."
  - "In three-dimensional space, the dot product of two vectors is given by the formula: u · v = ux*vx + uy*vy + uz*vz."

• noun

  In mathematics, the dot product (also called scalar product or inner product) is a binary operation that takes two equal-length sequences of numbers and returns a single number.

  - "The dot product of vectors u and v is calculated as the sum of the products of corresponding components."
  - "The dot product of two matrices A and B is defined as the sum of the products of the corresponding elements in each row of A and each column of B."

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Variants

List of all variants of dot product that leads to same result
dot product , dot products , inner product , scalar product , scalar products

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Etymology

origin and the way in which meanings have changed throughout history.

The term 'dot product' comes from the way the operation is represented when vectors are denoted as sequences of numbers. Each number in a vector is multiplied with the corresponding number in another vector, and the results are added together and represented as a single dot on a diagram.

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Trivia

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1. The dot product is sometimes called the scalar product because it produces a scalar (single number) result.

2. The dot product of two orthogonal (perpendicular) vectors is always zero.

3. The dot product of a vector with itself is called the square of the vector's magnitude.

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Related Concepts

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1. inner product: The dot product is a special case of an inner product, which is a more general concept that includes complex and non-real vector spaces.

2. matrix multiplication: The dot product of two matrices can be thought of as a special case of matrix multiplication, where one matrix is transformed into a vector and then dotted with another matrix.

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Culture

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The dot product is a fundamental concept in linear algebra and vector calculus. It is used extensively in physics, engineering, and computer graphics. The dot product is also used in the inner product space, which is a vector space endowed with an inner product.

How to Memorize "dot product"

1. visualize
   

   - To visualize the dot product, imagine two vectors represented by arrows. Multiply the magnitudes of the components of each vector in the same position and add the results. The dot product is the magnitude of the resulting vector, which is the projection of one vector onto another.

2. associate
   

   - To memorize the dot product, associate it with the term 'dot multiplication' or 'scalar multiplication'. Remember that it is a way to find the product of two vectors, resulting in a scalar value.

3. mnemonics
   

   - Use the mnemonic 'Dots and Crosses' to remember the dot product. Imagine two vectors, each with dots representing the components. Multiply corresponding dots and add the results to get the dot product.

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