Lessons 45–46 · 2 lessons · A. A. Rahimqoriyev, M. A. To‘xtaxo‘jayeva. Geometry, Grade 8 (textbook for general secondary schools), revised 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2014
45
Coordinates of a vector
Textbook: pp. 142–143
GoalKnow how to find the coordinates of a vector and write a vector by its coordinates.
New words
basis vectors · bazis vektorlarcoordinates of a vector · vektor koordinatalaribeginning of a vector · vektor boshiend of a vector · vektor oxiri
Explanation
In a Cartesian coordinate system xOy, the unit vector along the Ox axis is written →i and the unit vector along the Oy axis is →j; the pair (→i; →j) is called the basis vectors. A vector with its beginning at the origin O and end at A(x; y) is written →a = x→i + y→j, or briefly →a(x; y); x and y are the coordinates of the vector. If A₁(x₁; y₁) and A₂(x₂; y₂), the vector →A₁A₂ has coordinates (x₂ − x₁; y₂ − y₁): subtract the coordinates of the beginning from those of the end. If the beginning is A(x₁; y₁) and the coordinates are (a₁; a₂), the end is B(x₁ + a₁; y₁ + a₂).
Worked examples
A(2; 5), B(6; 8): →AB(6 − 2; 8 − 5) = →AB(4; 3).
The beginning is A(3; −1) and the vector is →a(−2; 4): the end is B(3 − 2; −1 + 4) = B(1; 3).
Class activity
“Vector on squared paper”: draw coordinate axes on squared paper, mark two points, compute the coordinates of the vector between them, then draw the arrow and check by counting squares.
Practice
1
A(−2; 3), B(4; 7). What is the first coordinate of →AB?
6
2
A(−2; 3), B(4; 7). What is the second coordinate of →AB?
4
3
The beginning is A(1; 5) and the coordinates are (3; −2). What is the second coordinate of the end?
3
4
Why are the coordinates of a vector that begins at the origin equal to the coordinates of its end?
The beginning has coordinates (0; 0), so x − 0 = x and y − 0 = y.