47
The scalar product of vectors
Textbook: pp. 146–147
GoalKnow the scalar product of vectors and the formula for the modulus of a vector, and use them.
New words
scalar product · skalar ko‘paytmascalar square · skalar kvadratmodulus of a vector · vektor modulicoordinates · koordinatalar
Explanation
The scalar product of →a(x₁; y₁) and →b(x₂; y₂) is the number x₁x₂ + y₁y₂: →a · →b = x₁x₂ + y₁y₂. The result is a number, not a vector. The scalar product of a vector with itself, →a · →a, is called the scalar square and written →a²; in coordinates →a² = x₁² + y₁². On the other hand →a² = |→a|², so |→a| = √(x₁² + y₁²). This formula for the length of a vector from its coordinates is really a consequence of the Pythagorean theorem: the coordinates are the legs of a right triangle. The distributive law over addition holds: (→a + →b) · →c = →a · →c + →b · →c.
Worked examples
→a(2; 3) · →b(4; 5) = 2 · 4 + 3 · 5 = 23.
For →a(9; 12), |→a| = √(81 + 144) = √225 = 15.
Class activity
“Length of a vector”: on squared paper draw an arrow 3 squares right and 4 squares up, measure its length and check that it equals the 5 given by the formula.
Practice
1
What is the scalar product →a(3; 4) · →b(2; 5)?
26
2
What is the modulus of the vector →a(8; 15)?
17
3
What is the scalar square →a² of the vector →a(6; 8)?
100
4
Why is |→a| = √(→a · →a)?
Because →a · →a = x² + y² = |→a|², the square of the hypotenuse of a triangle with legs x and y.