Vectors, Operations, and the Dot Product

Basic Terminology A vector in the plane is a directed line segment. Consider vector AB A is called the initial point B is called the terminal point Magnitude: length of a vector, expressed as The sum of two vectors is also a vector. The vector sum A + B is called the resultant.

Vectors

Basic Terminology continued A vector with its initial point at the origin is called a position vector. A position vector u with its endpoint at the point (a, b) is written The numbers a and b are the horizontal component and vertical component of vector u. The positive angle between the x-axis and a position vector is the direction angle for the vector.

Vector <a,b>

Magnitude and Direction Angle of Vector The magnitude (length) of vector u = is given by The direction angle  satisfies where a  0.

Example: Magnitude & Direction Find the magnitude and direction angle for Magnitude: Direction Angle: Vector u has a positive horizontal component. Vector u has a negative vertical component, placing the vector in quadrant ?.

Example: Magnitude & Direction Find the magnitude and direction angle for Magnitude: Direction Angle: Vector u has a positive horizontal component. Vector u has a negative vertical component, placing the vector in quadrant IV.

Horizontal and Vertical Components The horizontal and vertical components, respectively, of a vector u having magnitude |u| and direction angle  are given by That is,

Example: Components Vector w has magnitude 35.0 and direction angle 51.2. Find the horizontal and vertical components.

Example: Components Vector w has magnitude 35.0 and direction angle 51.2. Find the horizontal and vertical components. Therefore, w = The horizontal component is 21.9, and the vertical component is 27.3.

Example Write each vector in Figure 29 in the form

Example Write each vector in Figure 29 in the form

Solutions

Vector Operations For any real numbers a, b, c, d, and k,

Example: Vector Operations a) 4v b) 2u + v c) 2u  3v

i, j Forms for Vectors A unit vector is a vector that has magnitude 1. Dot Product The dot product of two vectors is denoted u • v, read “u dot v,” and given by

Example: Dot Product Find each dot product.

Properties of the Dot Product For all vectors u, v, and w and real numbers k, a) b) c) d) e) f)

Geometric Interpretation of Dot Product If  is the angle between the two nonzero vectors u and v, where 0   180, then

Example: Angle Between Find the angle  between the two vectors By the geometric

u + v

7.5 Applications of Vectors

Equilabrant

Example Forces of 10 newtons and 50 newtons act on an object at right angles to each other. Find the magnitude of the resultant and the angle of the resultant makes with the larger force. 10 50  v

Example Forces of 10 newtons and 50 newtons act on an object at right angles to each other. Find the magnitude of the resultant and the angle of the resultant makes with the larger force. The resultant vector, v, has magnitude 51 and make an angle of 11.3 with the larger force. 10 50  v

Example A vector w has a magnitude of 45 and rests on an incline of 20. Resolve the vector into its horizontal and vertical components. v u 45 20

Example A vector w has a magnitude of 45 and rests on an incline of 20. Resolve the vector into its horizontal and vertical components. The horizontal component is 42.3 and the vertical component is 15.4. v u 45 20

Example A ship leaves port on a bearing of 28.0 and travels 8.20 mi. The ship then turns due east and travels 4.30 mi. How far is the ship from port? What is its bearing from port?

Example continued Vectors PA and AE represent the ship’s path. Magnitude and bearing:

Example continued The ship is about 10.9 mi from port. To find the bearing of the ship from port, find angle APE. Add 20.4 to 28.0 to find that the bearing is 48.4.

Vector Problem