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Vector Dot Product

Vector Dot Product

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Multiplication of Vectors


 »  Two products because of orthogonality of components of vectors
    →  dot product is defined for components in parallel `vec p cdot vec q` `=vec p cdot vec a`
    →  cross product is defined for components in perpendicular `vec p xx vec q` `=vec p xx vec b`

Multiplication of Vectors

plain and simple summary

nub

plain and simple summary

nub

dummy

When two vector quantities interact to form a product, either one of the (1) component in parallel or (2) component in perpendicular is involved in the multiplication. In practical scenarios, when one component interacts, the other component does not interact and is lost in the product.

simple steps to build the foundation

trek

simple steps to build the foundation

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In this page, learn in detail about basics of 'vector multiplication'.


Keep tapping on the content to continue learning.
Starting on learning basics of "Multiplication of Vectors". ;; In this page, learn in detail about basics of vector multiplication.

What are the fundamental operations of mathematics?

  • Addition and Subtraction
  • Multiplication and Division
  • all the above

The answer is 'All the Above'. The four fundamental operations of mathematics are Addition, Subtraction, Multiplication, and Division.

We have studied about vector addition and multiplication of vectors by a scalar.

Is there something as vector multiplication?
That is the subject of learning in these pages. Let us go through some basics to understand vector multiplication.

    Consider the `vec p` and `vec q` shown in figure. `vec q` is split into `vec a` and `vec b`, such that
      `vec q = vec a + vec b` and
      `vec a` is in parallel to `vec p`
      `vec b` is in perpendicular to `vec p`
    vector multiplication When the vectors `vec p` and `vec q` interact as a product which component(s) of `vec q` take part in the multiplication?

    • only `vec a`
    • only `vec b`
    • either one : `vec a` or `vec b`
    • both together : `vec a` and `vec b`

    The answer is 'either one : `vec a` or `vec b`'. Remember the orthogonality of directions. When vectors interact to form products the interaction is either with the component in parallel or the component in perpendicular.

    Orthogonality of Directions : Any change in a direction affects component along that direction only and does not affect the components in the directions at `90^@` to that direction.

    The component which is in perpendicular to a direction does not affect the component in the direction. The directions which are at `90^@` angle are referred to as orthogonal directions.

    In arithmetic operations like addition, subtraction, multiplication by a scalar, and product of vectors, – the results are computed for

      •  components in parallel

      •  components in perpendicular

    Each of these two components interact differently.

    comprehensive information for quick review

    Jogger

    comprehensive information for quick review

    Jogger

    dummy

    How does direction affect vector quantities in multiplication? vector multiplication At an abstract level, there are two products possible. Given multiplicand `vec p` and multiplier `vec q`. `vec q` is split into `vec a` and `vec b`, such that
      `vec q = vec a + vec b` and
      `vec a` is in parallel to `vec p`
      `vec b` is in perpendicular to `vec p`

    two forms of multiplications are defined for each of these two components.

     •  one with component in parallel to the other, called vector dot product.`vec p cdot vec q = vec p cdot vec a`

     •  another with component in perpendicular to the vector, called vector cross product. `vec p times vec q = vec p times vec b`



               

    practice questions to master the knowledge

    Exercise

    practice questions to master the knowledge

    Exercise

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    What are the fundamental operations of mathematics?
    addition;subtraction
    Addition and Subtraction
    multiplication;division
    Multiplication and Division
    all;above
    all the above
    The answer is 'All the Above'. The four fundamental operations of mathematics are Addition, Subtraction, Multiplication, and Division.
    We have studied about vector addition and multiplication of vectors by a scalar.;; Is there something as vector multiplication? ; That is the subject of learning in these pages. Let us go through some basics to understand vector multiplication.
    Consider the vector p and vector q shown in figure. Vector q is split into vector a and vector b, such that ;; vector q = vector a + vector b and ;; vector a is in parallel to vector p ;; vector b is in perpendicular to vector p. ;; When vectors p and q interact as a product which component or components of vector q take part in the multiplication?
    1
    2
    3
    4
    The answer is "Either one : vector a or vector b". Remember the orthogonality of directions. When vectors interact to form products the interactionis either with the component in parallel or the component in perpendicular.
    Orthogonality of Directions : Any change in a direction affects component along that direction only and does not affect the components in the directions at 90 degree to that direction. ;; The component which is in perpendicular to a direction does not affect the component in the direction. The directions which are at 90 degree angle are referred to as orthogonal directions.
    In arithmetic operations like addition, subtraction, multiplication by a scalar, and product of vectors, – the results are computed for ;; components in parallel ;; components in perpendicular ;; Each of these two components interact differently.
    When two vector quantities interact to form a product, either one of the ;; component in parallel or ;; component in perpendicular is involved in the multiplication. In practical scenarios, when one component interacts, the other component does not interact and is lost in the product.
    How does direction affect vector quantities in multiplication? ;; At an abstract level, there are two products possible. Multiplicand vector p and multiplier vector q are given. ;; Vector q is split into vector a and vector b, such that vector q = vector a + vector b and ;; vector a is in parallel to vector p ;; vector b is in perpendicular to vector p. ;; two forms of multiplications are defined for each of these two components. ;; one with component in parallel to the other, called vector dot product. vector p dot vector q = vector p dot vector a ;; another with component in perpendicular to the vector, called vector cross product. vector p cross vector q = vector p cross vector b

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