Vectors

Basic concepts

A vector is a directed line segment. Its coordinates are written as a = (x; y).

• Vector length |a|: length: √(x² + y²). In space, |a| = √(x² + y² + z²).
• The vector from A(x₁; y₁) to B(x₂; y₂): AB = (x₂ − x₁; y₂ − y₁). That is, the end point minus the starting point.
• Midpoint of a segment: M((x₁ + x₂)/2; (y₁ + y₂)/2).

Operations

Operation	In coordinates
Sum	(x₁ + x₂; y₁ + y₂)
Difference	(x₁ − x₂; y₁ − y₂)
Multiplication by a number	k · a = (kx; ky)

Dot product

• a · b = x₁x₂ + y₁y₂
• a · b = |a| · |b| · cos φ, hence cos φ = (a · b) / (|a| · |b|).
• if a · b = 0, the vectors are perpendicular (for nonzero vectors).
• Collinear (parallel) vectors: x₁ / x₂ = y₁ / y₂, that is, x₁y₂ = x₂y₁.
• a · a = |a|².

1. Find the vector's coordinates (end minus start)
2. Choose the required quantity: length, dot product, angle
3. Substitute the values into the formula
4. Use the sign of cos φ to check whether the angle is acute or obtuse

Examples

• If a = (−6; 8), then |a| = √(36 + 64) = 10.
• a = (2; −1), b = (3; 4): a · b = 6 − 4 = 2.
• a = (2; k), b = (3; −6) are perpendicular: 6 − 6k = 0, k = 1. If they are collinear, 2 / 3 = k / (−6), k = −4.
• a = (1; √3), b = (−2; 0): a · b = −2, |a| = 2, |b| = 2, cos φ = −1/2, φ = 120°.

Mistakes

• if cos φ is negative, the angle is obtuse; do not replace it with 60°.
• The condition for perpendicularity is that the dot product equals zero; the condition for collinearity is proportionality.