2.2Components of a Vector

Objective:Find the components of a vector.

Number plane, or Cartesian coordinate system – a plane determined by the horizontal line called the x-axis and a vertical line called the y-axis intersecting at right angles. Plane divided in two

four quadrants: I, II, III, and IV.

Vector drawingsA directed line segment; arrowhead indicates

directionLength indicates magnitudeStarts with initial pointEnds with terminal point, or end point

R

The sum of two or more vectors is called the resultant vector.

When two or more vectors are added are added, each vector is called a component vector.

Useful component vectors are the components that are both perpendicular and that are parallel to the x- and y-axes.The horizontal component is called the x-

component.The vertical component is called the y-

component.R

Rx

Ry

The x- and y-components can be expressed as signed numbers.Absolute value indicates magnitude of vector.The sign indicates the direction.

x-component y-component

+, if right +, if up

-, if left -, if down

Find the x- and y-components of vector R.

R

Find the x- and y-components of vector v.

v

Find the x- and y-components of vector u.

u

We can find the resultant of several vectors using arithmetic graphing.The sum of all x-components for each vector

gives the x-component for the resultant vector, Rx.

The sum of all y-components for each vector gives the y-component for the resultant vector, Ry.

A

BR

Ax = +3

Ay = +2

By = +2

By = +4

Vector x-comp

y-comp

A +3 +2

B +2 +4

R +5 +6

Given vectors A, B, and C, graph and find the x- and y-components of the resultant vector R.

A

C

B

Vector x-comp

y-comp

A

B

C

R

Two vectors are equal when they have the same magnitude and the same direction.

They are opposites or negatives of each other when they have the same magnitude but opposite direction

A vector may be placed in any position in the number plane as long as its magnitude and direction are not changed.

To add two or more vector graphically:1. Draw first vector with initial point at origin2. The second vector should be drawn with its

initial point on the on the end point of first vector

3. If there are more vectors repeat until the last one drawn.

4. The sum, or the resultant vector, is drawn by joining the initial point of the first vector to the end point of the last vector.

Note: the order in which the vector are added does not matter.

Given vectors A, B, and C, graph and find the x- and y- components of the resultant vector R.

A

B

C

AB

CR

Rx = +1Ry = -5