12 2 Vectors Definitions Quantities that have magnitude

12. 2 Vectors

Definitions Ø Quantities that have magnitude but not direction are called scalars. Ex: Area, volume, temperature, time, etc. Ø Quantities such as force, acceleration, velocity or displacement that have direction as well as magnitude are represented by directed line segments, called vectors. B terminal point initial point A Ø The length of the vector is called the magnitude and is denoted by

Ø Vectors are equivalent if they have the same length and direction. y Ø A vector is in standard position if the initial point is at the origin. Ø The component form of this vector is: x Ø If (c, d) P are initial and terminal points of a vector, then the component form of (a, b) Q is: v (a-c, b-d) x

Example (-3, 4) P (-5, 2) The component form of is: Q v (-2, -2) The magnitude is

The magnitude of is: If then v is a zero vector : If then v is a unit vector. and are called the standard basis vectors. j i

Vectors in Space The magnitude of If then v is a zero vector : If then v is a unit vector. is: and are called the standard basis vectors.

Vector Operations Vector sum: Scalar Multiplication: Negative (opposite): Vector difference Vector v is parallel to u if and only if v = ku for some k.

Parallelogram Law u v u+v v u u v u-v v u + v is the resultant vector. u u - v is the resultant vector.

Application A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70 -mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E

A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70 -mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N u E

A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70 -mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N v 60 o u E

A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70 -mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N We need to find the magnitude and direction of the resultant vector u + v. v u+v u E

The component forms of u and v are: N v 70 u+v 500 u Therefore: and: The new ground speed of the airplane is about 538. 4 mph, and its new direction is about 6. 5 o north of east. E

Linear Combination v Any vectors can be written uniquely in terms of standard basis vectors : v is called a linear combination of i and j j i If v is any nonzero vector that makes an angle (measured counterclockwise) with the positive x-axis then v can be written as

Linear Combination in Space Standard basis vector notation v is called a linear combination of i, j and k

Examples 1) Find the unit vector in the direction of v 2) Determine whether the points are collinear: 3) Show that the following points form the vertices of a parallelogram: