☰ Contents · Geometry

Coordinates of a vector and operations on them

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
46

Operations on vectors given by coordinates

Textbook: pp. 144–145
GoalKnow how to add, subtract and multiply by a number vectors given by their coordinates.
New words
adding by coordinates · koordinatalar bo‘yicha qo‘shishsubtracting by coordinates · koordinatalar bo‘yicha ayirishmultiplying by a number · songa ko‘paytirishequal vectors · teng vektorlar
Explanation

Let vectors be given by coordinates: →a(x₁; y₁) and →b(x₂; y₂). The sum →a + →b has coordinates (x₁ + x₂; y₁ + y₂) and the difference →a − →b has (x₁ − x₂; y₁ − y₂). The product k→a has coordinates (kx₁; ky₁): each coordinate is multiplied by k. These rules follow by writing the vectors with the basis vectors →i and →j and using the properties of addition and multiplication by a number. Two vectors are equal exactly when their corresponding coordinates are equal.

Worked examples
→a(2; −3), →b(4; 5): →a + →b = (2 + 4; −3 + 5) = (6; 2).
3→a − →b = (3 · 2 − 4; 3 · (−3) − 5) = (2; −14).
Class activity

“Coordinate relay”: think of the coordinates of two vectors; your partner finds their sum and difference, then you swap roles.

Practice
1
→a(3; 7), →b(5; 2). What is the first coordinate of →a + →b?
2
→a(3; 7), →b(5; 2). What is the second coordinate of 4→a − →b?
3
→a(1; 4), →b(3; 2). What is the second coordinate of 2→a + →b?
4
Why do equal vectors have equal corresponding coordinates?