The idea of a vector. Adding and subtracting vectors
Lessons 41–42 · 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
42
Adding and subtracting vectors
Textbook: pp. 132–135
GoalKnow and apply the rules for adding and subtracting vectors.
New words
triangle rule · uchburchak qoidasiparallelogram rule · parallelogramm qoidasiopposite vector · qarama-qarshi vektordifference of vectors · vektorlar ayirmasi
Explanation
Triangle rule: for any points A, B, C we have →AB + →BC = →AC; the second vector is placed at the end of the first. Parallelogram rule: place two vectors at a common beginning and build a parallelogram on them; the sum is the diagonal from that beginning. Addition is commutative and associative: →a + →b = →b + →a and (→a + →b) + →c = →a + (→b + →c). Two vectors whose sum is the zero vector are opposite vectors, and the vector opposite to →a is written −→a. The difference is →a − →b = →a + (−→b); for vectors placed at a common point O we have →OA − →OB = →BA.
Worked examples
A closed path: →AB + →BC + →CA is the zero vector, because we return to the starting point.
→AC − →AB = →BC: both vectors start at A, and the difference goes from the end of the second to the end of the first.
Class activity
“Sum of a journey”: mark three points on paper, draw arrows from A to B and from B to C, draw the sum arrow from A to C, then swap the order of two vectors and check that the result is the same.
Practice
1
Vectors →a and →b point the same way, |→a| = 7 and |→b| = 5. What is |→a + →b|?
12
2
Vectors →a and →b point opposite ways, |→a| = 7 and |→b| = 5. What is |→a + →b|?
2
3
Which vector is equal to →AB + →BC + →CD?
→AD
4
Why is →AB + →BA the zero vector?
These are opposite vectors: we go from A to B and come back to A.