Vectors

Vectors Vectors describe magnitude and direction of a quantity. A vector can be represented by a directed line segment with an initial point and a terminal point. Vectors can be written as

Vectors The magnitude of , denoted , is the length of the vector from its initial point to its terminal point. The direction of a vector can be expressed as the angle it forms with the horizontal or as a measurement between 0 and 90 east or west of the north-south line.

Resultant A resultant is the sum of two or more vectors.

Example Copy the vectors. Then find .

Example Copy the vectors. Then find .

Vectors on the Coordinate Plane A vector is in standard position if its initial point is at the origin and can be described by its terminal point P(x, y). To describe a vector with any initial point, you can use the component form ⟨x, y⟩, which describes the vector in terms of its horizontal component x and vertical component y. To write the component form of a vector with initial point (x1, y1) and terminal point (x2, y2), find ⟨x2 – x1, y2 – y1⟩

Examples Write the component form of vector CD.

Examples Write the component form of vector PQ.

Magnitude and Direction The magnitude of a vector can be found by using the Distance Formula, and the direction can be found by using trigonometric ratios.

Examples Find the magnitude and direction of = ⟨-1,4⟩

Examples Find the magnitude and direction of = ⟨-1,4⟩ Magnitude: or about 4.12

Examples Find the magnitude and direction of = ⟨-1,4⟩ Direction Graph and its components. The angle is the inverse tangent of the components. θ = tan-1(1/4); θ = 14˚ The direction of is the angle that it makes with the positive x-axis, which is about 104˚.

Examples

Vector Operations

Examples Find for , , and .

Examples Find for , , and . ⟨1 - 3, -2 - 4⟩ ⟨-2, -6⟩