Vector Quantities We will concern ourselves with two

measurable quantities: Scalar quantities: physical

quantities expressed in terms of a magnitude only. Scalar quantities consist of a number and a unit. Example: 100 m/s (no direction).

Vector Quantities Vector quantities: quantities that are

described by a magnitude and a direction. Vector quantities consist of a number, a unit, and a direction (basically, a scalar quantity with the direction indicated). Example: if east is considered to be positive, then west is negative, so the vector can described as 100 m/s, east or +100 m/s; 80 m/s, west or -80 m/s.

Vector Quantities Vectors are represented by an arrow

(). The tail of the arrow will always be placed at the point of origin for the measurement of the desired quantity.

The length of the arrow indicates the magnitude of the vector.

The orientation of the arrow indicates the direction.

Vector Resolution Resultant: is the single vector that

produces the same result as the combination of the separate vectors. Each separate vector is called a component vector.

We will examine the impact of all vector quantities acting upon an object as if all the vectors originate at a single point.

Vector Addition

1. When two or more vectors act at the same point in the same direction (in other words, the angle between each of the vectors is 0), the resultant is determined by adding all the component vectors together.

2. The direction of the resultant vector is in the direction of the component vectors.

Vector Addition If you walk 5 m to the

right, stop, and then walk 3 m to the right, the total displacement is: 5 m + 3 m = 8 m.

The 5 m vector and the 3 m vector are the component vectors.

Resultant displacement = 8 m to the right

Properties of Vectors (HRW)

Vector Subtraction 1. When two vectors act at the same

point, but in opposite directions (the angle between each vector is 180), the resultant is equal to the difference between the two vector quantities (subtract).

2. The direction of the resultant is in the direction of the component vector with the largest magnitude.

Vector Subtraction If you walk 5 m to the

right, stop, and then walk 3 m to the left, the total displacement is:

5 m – 3 m = 2 m The 5 m vector and the 3

m vector are the component vectors.

Resultant displacement = 2 m to the right.

Subtraction of Vectors (HRW)

Head to Tail Method When two or more vectors

act at the same time (concurrently) on the same point, the resultant can be determined by placing the vectors head to tail.

The tail of the first vector (A) will begin at the origin and the tail of the next component vector (B) is placed at the head of the vector A.

Head to Tail Method The tail of vector C is

placed at the head of vector B.

Each component vector is drawn with the correct orientation.

The order in which the vectors are drawn does not matter as long as the magnitude and direction of each vector is maintained when drawn.

Head to Tail Method The resultant vector

R would be the vector beginning at the origin and extending in a straight line to the head of the last vector. The determination of the magnitude will involve use of the Pythagorean theorem or the law of cosines.

Head to Tail Method The direction of the resultant can be

determined by resolving the resultant vector into x (horizontal) and y (vertical) components and using a trig function (sin, cos, or tan).

This will be described later. Graphical Addition of Vectors & Adding

Vectors Algebraically (HRW)

For Vector Problems Involving Right Angles Example: move 5 m

along the x-axis and then move 8 m up the y-axis.

Place the origin of a coordinate system at the point where the motion begins or where the force is applied; you will start at (0,0) on the coordinate system.

For Vector Problems Involving Right Angles

Tail of the 5 m vector goes at the origin and the head points along the x-axis and would lie on 5 on the axis.

At the head of the 5 m vector, turn up and the tail of the 8 m vector will begin and the head points up the y-axis and would lie on 8 on the axis.

The head of the 8 m vector would lie on the point (5,8) on an x-y coordinate plane.

For Vector Problems Involving Right Angles Resultant vector R

begins at the origin (0,0) and ends at the head of the 8 m vector [at the point (8,5)].

The 5 m vector and the 8 m vector are the component vectors.

For Vector Problems Involving Right Angles The resultant R is the

hypotenuse of a right triangle formed by the 5 m vector and the 8 m vector.

Use the Pythagorean theorem to determine the magnitude of the resultant R: R2 = A2 + B2

R2 = (5 m)2 + (8 m)2

R2 = 89 m2; R = 9.43 m

For Vector Problems Involving Right Angles

To determine the direction of the resultant vector: Requires an angular measurement and a

direction moved from a reference axis. Example: 40 above the x-axis.

Choose one of the two angles at the point of origin. The reference axis is the adjacent side of the selected angle.

For Vector Problems Involving Right Angles

Trig Functions Any one of the trig functions (sin, cos, or tan) can be used to find the direction of the resultant vector R.

Be sure the calculator is in degree mode!

hypotenuseadjacent

θcos

hypotenuseopposite

θsin

adjacentopposite

θtan

1

1

1

sin

cos

tan

oppositehypotenuse

adjacenthypotenuse

oppositeadjacent

For Vector Problems Involving Right Angles

Report the angle determined with the trig function and the direction moved from the reference axis. Example:

Use the inverse tan function (tan-1) to determine the angle

6.1m5m8

θtan

For Vector Problems Involving Right Angles

= tan-1 1.6 = 57.995o

To get to the resultant, you must start at the x-axis and rotate 57.995o above the x-axis.

The magnitude of the resultant is 9.43 m and the direction is 57.995o above the x-axis.

X and Y Components of a Vector

Vectors can be resolved (broken down) into a component that acts along the x-axis and a component that acts along the y-axis.

The tail of the vector to be resolved is placed at the origin and drawn as indicated in the problem.

X and Y Components of a Vector Ex: the 50 m/s

vector located 30° above the positive x-axis will be resolved into an x-component and a y-component.

X and Y Components of a Vector

From the origin, draw a line along the x-axis to a point below the tip of the head of the vector (the arrow head). This is the x-component of the vector.

From the origin, draw a line along the y-axis to a point adjacent to the tip of the head of the vector (the arrow head). This is the y-component of the vector.

X and Y Components of a Vector

Construct a parallelogram (either a square or a rectangle). A parallelogram is used because the opposite sides of a parallelogram are equal in magnitude. You also have two right triangles which you can use to solve for the components.

The 50 m/s the diagonal of the parallelogram and will be the hypotenuse for the right triangle you will use to determine the x-component and the y-component.

X and Y Components of a Vector The x-component

is the adjacent side of the right triangle and you will use the cosine function to determine its magnitude.

X and Y Components of a Vector

s/m3.43x

30coss/m50x

multiplycross

s/m50x

30cos

hypadj

θcos

X and Y Components of a Vector The y-component

is the opposite side of the right triangle and you will use the sine function to determine its magnitude.

X and Y Components of a Vector

smy

smy

multiplycross

smy

hypopp

/25

30sin/50

/5030sin

sin

X and Y Components of a Vector Summarized

For problems involving multiple vectors: Rectangular Resolution Resolve each vector into an x-

component and a y-component using the trig functions sine or cosine, whichever is appropriate. Be careful to denote the negative values for the x- and y-components, when appropriate.

Add all the x-components together to get one resultant x-component vector.

Add all the y-components together to get one resultant y-component vector.

For problems involving multiple vectors: Rectangular Resolution

Use the Pythagorean theorem to determine the resultant vector.

Use a trig function (sine, cosine, or tangent) to determine the angle of orientation for the resultant vector.

Visit Adding Vectors Algebraically (HRW)

Example 3C, p. 95 Given two vectors:

25.5 km at 35° south of east 41 km at 65° north of east

Draw the two vectors on the coordinate grid. The tail of the first vector goes at the origin.

Example 3C, p. 95 To draw 35° south of

east: place the protractor on the origin with the 90° mark on the south axis and the 0° mark on the east axis. Measure 35° from the east axis toward the south axis. Draw the 25.5 km vector from the origin at 35°.

Example 3C, p. 95 Make the head of the 25.5 km vector

the origin. From the head of the 25.5 km vector, draw the 41 km vector with its tail at the origin and measure the 65° angle from the horizontal axis.

To draw 65° north of east: place the protractor on the origin with the 90° mark on the north axis and the 0° mark on the east axis. Measure 65° from the east axis toward the north axis. Draw the 41 km vector from the origin at 65°.

Example 3C, p. 95

Example 3C, p. 95 The resultant

vector R is the vector that begins at the origin and ends at the head of the 41 km vector.

Determine the x-component and the y-component for the two vectors.

Example 3C, p. 95

Example 3C, p. 95 25.5 km: x-component

km89.20x

35coskm5.25x

km5.25x

35cos

1

1

1

Example 3C, p. 95 25.5 km: y-component

km63.14y

35sinkm5.25y

km5.25y

35sin

1

1

1

Example 3C, p. 95

Example 3C, p. 95 41 km: x-component

kmx

kmx

km

x

33.17

65cos41

4165cos

2

2

2

Example 3C, p. 95

kmy

kmy

km

y

16.37

65sin41

4165sin

2

2

2

41 km: y-component

Example 3C, p. 95 Place the two x-component vectors

and the two y-component vectors at the origin.

Examine each vector to determine if it should have a positive or negative sign based upon its direction on the x-y coordinate plane.

Example 3C, p. 95 Both x1 and x2 will

be positive. Y2 will be positive. Y1 will be negative. Because x1 and x2

are in the same direction, add them to get a single x-component vector.

Example 3C, p. 95 x1 + x2 = 20.89 km + 17.33 km =

38.22 km The x-component of the resultant

is 38.22 km. y1 and y2 are in opposite directions,

make sure they have the correct signs and add them to get a single y-component vector.

Example 3C, p. 95 y1 + y2 = -14.63 km + 37.16 km =

22.53 km The y-component of the resultant is

22.53 km. Place the tail of the x-component of the

resultant at the origin along the positive x-axis.

Place the tail of the y-component of the resultant at the origin along the positive y-axis.

Example 3C, p. 95

Example 3C, p. 95 Redraw the y-

component vector with its tail at the head of the x-component vector.

The tail of the resultant R begins at the origin and goes to the head of the y-component vector.

Example 3C, p. 95 Use a2 + b2 = R2 to determine R.

km35.44km84.1966R

Rkm84.1966

Rkm6.507km24.1459

Rkm53.22km2.38

2

22

222

222

Example 3C, p. 95 The magnitude of

the resultant vector is 44.35 km.

Use sin, cos, or tan to determine the direction of the resultant. This requires an angle measurement and the orientation of the vector.

Example 3C, p. 95

o

1

53.30θ

5898.0tanθ

5898.0km2.38km53.22

θtan

Example 3C, p. 95 The resultant vector is 30.53° north of the east

axis. In other words, to locate the resultant, you start at the east axis and measure an angle of 30.53° towards the north axis.

For determining the orientation of a vector, the angle is given first, then the direction of rotation is given and then the axis from which the rotation occurred is given.

An angle measurement of 59.47° east of the north axis is also correct when determining the location of the resultant vector.

Note: For orientations given as northwest, northeast, southeast, or southwest, θ = 45°.

Casao’s Wheel of Directions