43
Multiplying a vector by a number
Textbook: pp. 136–138
GoalKnow multiplication of a vector by a number and the idea of a unit vector.
New words
multiplying a vector by a number · vektorni songa ko‘paytirishunit vector · birlik vektorcollinearity · kollinearlikmodulus · modul
Explanation
The product of a non-zero vector →a by a number k is the vector k→a whose modulus is |k| · |→a|; its direction is the same as →a if k > 0 and opposite if k < 0. The product of the zero vector by any number, and of any vector by zero, is the zero vector. The product is always collinear with the given vector. Properties: (k · l)→a = k · (l→a), (k + l)→a = k→a + l→a, k(→a + →b) = k→a + k→b. Dividing a vector by its own modulus gives a unit vector with modulus 1: →e = →a : |→a|, so →a = |→a| · →e.
Worked examples
|→a| = 5 and k = −3: |k→a| = 3 · 5 = 15, and k→a points opposite to →a.
If |→a| = 20, then →e = →a : 20 is a unit vector, |→e| = 1 and →a = 20→e.
Class activity
“Stretch and shrink”: draw an arrow →a in your notebook, then draw 2→a, 3→a and (−1)→a and compare their lengths and directions.
Practice
1
|→a| = 4 and k = 3. What is |k→a|?
12
2
|→a| = 6 and k = −5. What is |k→a|?
30
3
|→a| = 7 and →e is the unit vector codirectional with →a. →a = ? · →e
7
4
Why is (−1) · →a the vector opposite to →a?
Its modulus is |→a|, its direction is opposite, and the sum with →a is the zero vector.