3.3 Vectors in the Plane

Numbers that need both magnitude and direction to be described are called vectors.

To represent such a quantity we use a directed line segment.

Initial Point

Terminal Point

Notation• Vectors are written as arrows. – The length of the arrow describes the magnitude

of the vector.– The direction of the arrow indicates the direction

of the vector…• Vectors are written in bold in your book• On the board we will use the notation below…

PQ u, v, or w

• You can think of magnitude as size or amount, including units.

To find the magnitude we use the distance formula

EX– let u be represented by the directed line segment from P(0, 0) to Q (3,2) and let v be represented by the directed line segment from R (1,2) to S (4,4). Graph these vectors and find the magnitude.

• Component form of the vector with initial point P = (p1, p2) and terminal point Q = (q1, q2) is given by

PQ = < q1- p1, q2- p2> = <v1, v2> = v

The magnitude (or length) of v is given by

||v|| = (q1,-p1)2 + (q2,-p2)2

If ||v|| = 1 then it is called the unit vector

Example

• Find the component form and the magnitude of the vectors with initial point (1, 11) and terminal point (9,3)

• Scalar multiplication• Vector addition

• In operations with vectors, numbers are usually referred to as scalars.

• The resultant is the sum of two or more vectors added together.

Adding vectors

• u + v = < u1+v1, u2+ v2> • Graphically--

• Example=Let v = <-2, 5> and w= <3,4>

• So v + w =

Scalar multiplication

• 2u = < 2u1, 2u2> • Graphically--

• Example=Let v = <-2, 5> and w= <3,4>

• So 2v =

Unit Vectors

• u = unit vector = v = 1 v ||v|| ||V||

• Find the unit vector in the direction of w = 7j – 3i

• We know a vector with the initial point at the origin is said to be in in standard position.

• u = <a, b>• Where a is the horizontal component and b

is the vertical component• NOW……• u = <a, b> AKA v = ai + bj• Where i is the horizontal component and j is

the vertical component

IF u = -3i + 8j and v = 2i - j

• Find 2u – 3v

• The positive angle between the x-axis and a positive vector

• How would you find this angle?

Find the direction angle of the vector u = 3i + 3j

Find the direction angle of the vector u = 3i - 4j

Applications of Vectors• 2 ways to write vectors

<a,b> or u = ai + bj• Each time we did this we made a graph….

<a,b>(x, y)(cos , sin )

u = cos i, sin j

Component Form

• v = ||v||(cos)i + ||v||sinj