4.8

Topic

Vectors

Practice by unit: Functions Involving Parameters, Vectors, and Matrices

Sub-standards

• 4.8.A.1 — Identify characteristics of a vector. A vector is a directed line segment. When a vector is placed in the plane, the point at the beginning of the line segment is called the tail, and the point at the end of the line segment is called the head. The length of the line segment is the magnitude of the vector.
• 4.8.A.2 — A vector _ → ___ P P1 2 with two components can be plotted in the xy-plane from P = (x , y1 ) 1 to 1 P = (x , y2 ) 2 2 . The vector is identified by a and b, where a = x − x1 and b = y − y1. 2 2 The vector can be expressed as a b, . A zero vector 0, 0 is the trivial case when P = P . 1 2
• 4.8.A.3 — The direction of the vector is parallel to the line segment from the origin to the point with coordinates (a b, ). The magnitude of the vector is the square root of the sum of the squares of the components.
• 4.8.A.4 — For a vector represented geometrically in the plane, the components of the vector can be found using trigonometry.
• 4.8.B.1 — Determine sums and products involving vectors. The multiplication of a constant and a vector results in a new vector whose components are found by multiplying the constant by each of the components of the original vector. The new vector is parallel to the original vector. UNIT 4 a b, a b, = a a 1 2 . 1 1 2 2 1 2 + b b a b →, i = 1, 0 and j = 0, 1 .
• 4.8.B.3 — The dot product of two vectors is the sum of the products of their corresponding components. That is,
• 4.8.C.1 — Determine a unit vector for a given vector. A unit vector is a vector of magnitude 1. A unit vector in the same direction as a given nonzero vector can be found by scalar multiplying the vector by the reciprocal of its magnitude. →→ ai b + j
• 4.8.C.2 — → → → The vector can be expressed as in R 2 , where i j and are unit vectors in the x and y directions, respectively. That is,