Lesson Properties of Dot-product of vectors in a coordinate plane

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Properties of Dot-product of vectors in a coordinate plane

In this lesson you will learn about properties of dot-product of vectors in a coordinate plane.

This lesson is a continuation of the lessons

    - Introduction to dot-product,
    - Formula for Dot-product of vectors in a coordinate plane via the vectors components,
    - Dot-product of vectors in a coordinate plane and the angle between two vectors, and
    - Orthogonal vectors in a coordinate plane

that are under the current topic in this site.

Let u and v are any two vectors in a coordinate plane u = (a,b), v = (c,d), where a and b are components of the vector u, c and d are components of the vector v, and alpha

is the angle between the vectors u and u.

Recall that

- dot-product was defined in the lesson Introduction to dot-product as

(u,v) = |u|*|v|* cos%28alpha%29

;                                (1)

    - the formula was derived in the lesson Formula for Dot-product of vectors in a coordinate plane via the vectors components expressing dot-product of vectors
       via their components

              (u,v) = a*c + b*d;                                       (2)

    - the formula was derived in the lesson Dot-product of vectors in a coordinate plane and the angle between two vectors expressing cosines of the angle between two non-zero
       vectors via their components:

               cos%28alpha%29

%28a%2Ac+%2B+b%2Ad%29%2F%28sqrt%28a%5E2%2Bb%5E2%29%2Asqrt%28c%5E2%2Bd%5E2%29%29

;                  (3)

    - it was established in the lesson Perpendicular vectors in a coordinate plane that the two vectors u = (a,b) and v = (c,d) are perpendicular if and only if the equality is held

              a*c + b*d = 0                                                (4)

       for the vectors' components.

Properties of scalar product

1. If vectors u and v are given by their component form u = (a,b) and v = (c,d) then (u,v) = a*c + b*d.

2. (u,v) = (v,u)                         (commutativity).

3. (u,v + w) = (u,v) + (u,w)      (distributivity to addition of vectors).

4. If vectors u and v are parallel and have the same direction (are co-directed) then (u,v) = |u|*|v|.
     If vectors u and v are parallel and have the opposite directions then (u,v) = -|u|*|v|.

5. (ru,v) = r*(u,v)                   (associativity with respect to scalar factor r).

6. (u,u) = abs%28u%29%5E2

(scalar product of a vector u by itself is equal to the square of the length of the vector, or, which is the same,
|u| = sqrt%28a%5E2+%2B+b%5E2%29

                     the length of a vector is equal to the square root of the scalar product of the vector by itself, where a and b are the vector's components).

7. Two non-zero vectors u and v are perpendicular if and only if their scalar product is equal to zero: (u,v) = a*c + b*d = 0.

These properties follow the formulas (1), (2), (3) and (4).

My lessons on Dot-product in this site are
    - Introduction to dot-product
    - Formula for Dot-product of vectors in a plane via the vectors components
    - Dot-product of vectors in a coordinate plane and the angle between two vectors
    - Perpendicular vectors in a coordinate plane
    - Solved problems on Dot-product of vectors and the angle between two vectors
    - Properties of Dot-product of vectors in a coordinate plane                                                      (this lesson)
    - The formula for the angle between two vectors and the formula for cosines of the difference of two angles

There are short lessons of the "HOW TO . . . " type on Dot-product:
    - HOW TO find the length of the vector in a coordinate plane
    - HOW TO find dot-product of two vectors in a plane
    - HOW TO find scalar product of two vectors in a coordinate plane
    - HOW TO find the angle between two vectors in a coordinate plane
    - HOW TO prove that two vectors in a coordinate plane are perpendicular
    - HOW TO prove that a triangle in a coordinate plane is a right triangle
    - HOW TO check if a quadrilateral in a coordinate plane is a parallelogram
    - HOW TO check if a quadrilateral in a coordinate plane is a rectangle
    - HOW TO check if a quadrilateral in a coordinate plane is a rhombus
    - HOW TO check if a quadrilateral in a coordinate plane is a square

For the full list of my lessons on dot-product with short annotations see the file OVERVIEW of lessons on Dot-product.

Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.

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