Vector addition, subtraction

Fundamentals of 2-D vector addition, subtraction

WHAT CHANGES BETWEEN 1 AND 2 DIMENSIONAL PROBLEMS

2 Dimensional Motion, Forces

• Combining displacements or forces in 2 dimensions is not as straightforward as we have previously done– Not just simple addition or subtraction

• The direction for motion in 2 dimensions cannot be described by a simple plus or minus

Description

Boy is walking 25 meters in the North East Direction

Displacement value is =

25m [NE] or 25m (45°)

2D values

• Always written in the formMagnitude, direction

• Several ways to write the direction

[ ] use some combination of North, South, East West and the angle of direction

A single angle measured from the horizontal axis pointing right (Polar coordinates)

The description for force and motion values get more complicated (longer)

WHAT IS A VECTOR?

Vectors

• Arrows that visually represent velocity, force, displacement, and acceleration

• Indicates direction and size

• Writing vectors: 54N [E]

Step Back a bit…

• Displacement, Velocity, Acceleration, Forces… can all be represented by vectors

• We have already used vectors in FBD

DETERMINING the DIRECTION value

Direction using N,S,W,and E

• Use brackets

• First letter is the cardinal direction line used as the start of the measurement of the angle

• Number the value of the angle measured

• Last letter is the cardinal direction towards which you measure (never on the same dimension)

Coming up with the direction

Started with the south direction

Moving towards the east direction

34°

[S 34 E]

Polar Coordinates

• Angle is from 0 to 360 degrees

• 0 and 360° point due east (to the right)

• Measured going counterclockwise

Problems

Angle Descriptors:

[W 75 S]

[ S 15 W]

(255°)60°

75°Angle descriptors:

[W 60 N]

[N 30 E]

(120°)

Problem

• Describe the direction of the following vector. Come up with at least 4 correct values

13°

Answer

• [ E 13 S]

• [S 77 E]

• (347°)

• (-13°) rarely used

Drawing 2-D vectors based on names

• West of north

• [N 30 W]

HOW TO COMBINE VECTOR VALUES

Resultant Vector

• The vector value obtained when 2 or more vectors are combined (added or subtracted)

One Dimensional vector combinations

• 5 m [N] + 7 m [N] = ___________

• 5 m [N] + 7 m [S] = ___________

• 5 m [N] – 7 m [N] = ___________

• 5 m [N] – 7 m [S] = ___________

Combining Vector values in a single dimension

• Add if going the same direction

• Subtract if going opposite directions

Basics

• Combine all forces in the same dimensions

• Draw the vectors to form 2 sides of a right triangle (head to tail).

• Does not matter which one you will start with. Will determine angle write-up

• Use Pythagorean theorem to solve for side

• Use inverse trig to find angle

Combination of 2 vectors not in the same dimension

• 5 m [N] + 7 m [E] = _____________

5 m [N]

7 m [E]

The resultant vector

Use Pythagorean theorem to solve for the 3rd side,

and

tan-1 to find the angle

Answer

• To find the side:

52 + 72 = resultant2 = 8.9

To find the angle tan-1 (7/5) = 54.5°

Vector description: 8.9 m [ N 54.5 E]

Problem #1

• Combine

53 m/s [S] + 67m/s [W] + 127 m/s [N]

Answer

• First, combine 53 m/s [S] and 127 m/s [N]

Subtract small from large since they go in opposite directions

127 – 53 = 74 m/s [N]

(use direction of larger value)

Answer continued

• Next combine the sum of the above and 67 m/s [W]

74

67

Resultant

Answer, continued

• Use Pythagorean theorem to solve for the hypotenuse

• Use tan-1 (inverse tangent) to solve for the angle

Final answer : 99.8 m/s [W 47.8 N]

Class Problem

• A 110 N force and a 55 N force both act on an object at point P. The 55 N force acts at 0%. What is the magnitude and direction of the resultant force?

Point p55 N

110 N

Combining vectors in 2-dimensions

• Resultant vector: The vector that represents the sum of 2 or more vector values

• Drawing the resultant: Draw one vector. From the end of the first, draw the second.

• The resultant is the arrow drawn from the beginning of the first vector to the end of the last vector

Adding vectors in 2 dimensions

• If dimensions are perpendicular:

• Draw given vectors head to tail

• The resultant is drawn from the beginning of the first vector to the end of the last vector.

Solving for length of resultant vector

Pythagorean theorem

DETERMINING THE “OVERALL” VALUE

However

• If an overall value is being sought… answer will be 2-D

• Overall velocity

• Overall Displacement

• Overall Net Force

Overall Velocity

• If the overall velocity of the boat is wanted, then the velocity of the boat and the water must be combined to form the resultant vector

• We next figure out how to solve for the resultant

Steps

• Use Pythagorean theorem or trig to find the length of the hypotenuse (resultant vector)

• Use inverse trig to find the angle of inclination

• The angle to be solved for is between the initial side drawn and the hypotenuse!

SUBTRACTION OF VECTORS

Relationship between addition and subtraction

• Subtraction is = to adding the opposite value

Subtract

• 51.6 m [N] – 45.7 [W]