A vector quantity has magnitude and direction; a scalar quantity has magnitude only. (p.212)
Distance (scalar) vs displacement (vector); speed (scalar) vs velocity (vector); mass (scalar) vs weight (vector). (p.213)
A vector is represented by a directed line segment; notation AB→, a; magnitude |AB→| or |a|. (p.214)
Equal vectors have the same magnitude and direction. A negative vector has the same magnitude but opposite direction (BA→ = −AB→). A zero vector has zero magnitude. (p.215)
Scalar multiplication: ka has magnitude |k||a|, same direction if k > 0 and opposite if k < 0. (p.218)
Two vectors are parallel if and only if a = kb for a constant k. (p.219)
If a and b are non-zero and not parallel, and ha = kb, then h = k = 0. (p.219)
Points A, B, C are collinear if AB→ = kBC→ (parallel with a common point). (p.219)
8.2 Addition and Subtraction of Vectors
A resultant vector is the single vector produced by adding several vectors. (p.221)
Triangle law: AB→ + BC→ = AC→. Subtraction: a − b = a + (−b). (p.222)
Parallelogram law: the resultant of two vectors from the same point is the diagonal of the parallelogram. (p.222)
Polygon law: AB→ + BC→ + CD→ + DE→ = AE→. (p.222)
To show collinearity, write AC→ = kAB→ and compare coefficients of a and b. (p.225)
8.3 Vectors in a Cartesian Plane
Unit vectors: i = (1, 0) along the x-axis and j = (0, 1) along the y-axis, each of magnitude 1. (p.227)
Position vector OB→ of B(x, y) = xi + yj = column vector (x, y); magnitude = √(x2 + y2). (p.228)
Unit vector in the direction of r = xi + yj is r̂ = r / |r| = (xi + yj) / √(x2 + y2). (p.230)
A unit vector has magnitude 1. (p.231)
Addition/subtraction: add or subtract the i and j components separately. (p.232)
Multiplying by a scalar multiplies both components; do scalar multiplication before addition/subtraction. (p.233)
Moving particle: position s = a + tv, where a is the initial position vector and v the velocity vector; speed = |v|. (p.234)