6.1 – Vectors in the Plane

What are Vectors? Vectors are a quantity that have both magnitude (length) and direction, usually represented with an arrow: This includes force, velocity, and acceleration Component Form: v =

Naming Vectors A vector can also be written as the letters of its head and tail with an arrow above: A – initial point B – terminal point

Scalars A quantity with magnitude alone, but no directions, is not a vector, it’s called a scalar For example, the quantity “60 miles per hours” is a regular number, or scalar. The quantity “60 miles per hour to the northwest” is a vector, because it has both size and direction

Components To do computations with vectors, we place them in the plane and find their components. v (2,2) (5,6)

Components The initial point is the tail, the head is the terminal point. The components are obtained by subtracting coordinates of the initial point from those of the terminal point. v (2,2) (5,6)

Components The first component of v is 5 -2 = 3. The second is 6 -2 = 4. We write v = v (2,2) (5,6)

Magnitude of a Vector The magnitude (or length) of a vector is shown by two vertical bars on either side of the vector: |a| OR it can be written with double vertical bars: ||a||

Magnitude of a Vector Find the magnitude of the vector: V =

Finding Magnitude of a Vector

Showing Vectors are Equal Let u be the vector represented by the directed line segment from R to S, and v the vector represented by the directed line segment from O to P. Prove that u =v.

Addition To add vectors, simply add their components. For example, if v = and w =, then v + w =.

Multiples of Vectors Given a real number c, we can multiply a vector by c by multiplying its magnitude by c: v 2v2v -2v Notice that multiplying a vector by a negative real number reverses the direction.

Scalar Multiplication To multiply a vector by a real number, simply multiply each component by that number. If v = and w =, then: -2v = 4v – 2w =

Vector Operations Example

Unit Vectors A unit vector is a vector with magnitude (length) of 1. Given a vector v, we can form a unit vector by multiplying the vector by 1/||v||. Or you can think of this as v/||v||  (The vector divided by its magnitude)

Finding a Unit Vector

Standard Unit Vectors A vector such as can be written as For this reason, these vectors are given special names: i = and j =. A vector in component form v = can be written ai + bj. For example, rewrite the vector

Direction Angles The precise way to specify the direction of a vector is to state its direction angle (not its slope). v

Direction Angles

Finding the components of a Vector

Examples Find the component form of v, with magnitude 15 and a direction angle of 40 degrees. Find the component form of vector v with magnitude 6 and direction angle of 115 degrees.

Examples Find the component form of v, with magnitude 15 and a direction angle of 40 degrees. = Find the component form of vector v with magnitude 6 and direction angle of 115 degrees. =

Finding the Direction Angle of a Vector

Finding the direction angle Find the direction angle for the vector v

Velocity and Speed The velocity of a moving object is a vector because velocity has both magnitude and direction. The magnitude of velocity is speed.

Word Problem An airplane is flying on a compass heading (bearing) of 170 degrees at 460 mph. A wind is blowing with a bearing of 200 degrees at 80 mph. a) Find the component form of the velocity of the airplane b) Find the actual ground speed and direction of the plane.