31
The angle between two vectors and their dot product
Textbook: pp. 90–93
GoalKnow the angle between two vectors and the dot (scalar) product; compute the dot product, the length of a vector and the cosine of the angle in coordinates.
New words
dot (scalar) product · skalar ko‘paytmaangle between vectors · vektorlar orasidagi burchakperpendicular vectors · perpendikulyar vektorlarlength (magnitude) of a vector · vektor uzunligi
Explanation
If nonzero vectors a and b are placed at one point, the angle formed is the angle between them, φ. Their dot product is a·b = |a|·|b|·cosφ — a number, not a vector. If a·b = 0, the vectors are perpendicular, since cos90° = 0. In coordinates for a(a₁; a₂) and b(b₁; b₂): a·b = a₁b₁ + a₂b₂, |a| = √(a₁² + a₂²), and cosφ = (a₁b₁ + a₂b₂)/(|a|·|b|). Properties: a·b = b·a; (a + b)·c = a·c + b·c; a·a = |a|². In physics the work done by a force F moving a body through displacement s is A = F·s = |F|·|s|·cosφ.
Worked examples
For a(3; 4) and b(4; −3): a·b = 3·4 + 4·(−3) = 0, so a ⊥ b.
a(2; 0), b(1; 1): a·b = 2, |a| = 2, |b| = √2, cosφ = 2/(2√2) = √2/2, so φ = 45°.
Class activity
“Perpendicular?”: vector pairs are written on the board; students compute the dot product, find the perpendicular pair and check on a coordinate plane.
Practice
1
|a| = 6, |b| = 7 and the angle between them is 60°. What is a·b?
21
2
a(−1; 3) and b(2; 5). What is a·b?
13
3
For which x are a(2; x) and b(3; 6) perpendicular?
-1
4
Why is a·a = |a|²?
A vector makes a 0° angle with itself and cos0° = 1, so a·a = |a|·|a|·1.