VectorsAdding and Subtracting Vectors

Vectors

A vector quantity is a quantity that is fully described by both magnitude and direction. On the other hand, a scalar quantity is a quantity that is fully described by its magnitude.

ExampleVector: velocity = 30 mph north scalar: speed = 30 mph

30 mph North

Vector representation

Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction.

N

S

W E

Scale: 1 cm = 10 mph

3 cm = 30 mph

head

tail

Not so easy to represent vectors

Vectors can be directed due East, due West, due South, and due North. But some vectors are directed northeast (at a 45 degree angle); and some other vectors are even directed northeast, yet more north than east. This will cause a confusion in the exact direction. Thus, there is a clear need for some form of a convention for identifying the direction of a vector that is not due East, due West, due South, or due North.

N

S

W E

NE

NE

SE

NW

SW

Standard Angle Orientation

The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its "tail" from due East. Using this convention, a vector with a direction of 40 degrees is a vector that has been rotated degrees in a counterclockwise direction relative to due east.

40o

E

Positive x-axis

Angle Orientation

So where did all this come from. Geometry Unit Circle.

45o

45o

60o

30o

x

x

x√2 x√3

x

2x

45o - 45o - 90o

30o - 60o - 90o

Example

Draw a 60 m/s vector rotated counterclockwise 120o

Example

120o

Scale: 1 cm = 10 m/s

6 cm

0o

90o

180o

270o

Adding vectors 1-dimension

We can combine vectors to make a new a vector. The new vector is called the RESULTANT vector.

Simples rule for adding in one-dimension.

If they are going in the same direction, add up the magnitude (value) of the vectors.

+ =5 m/s 7 m/s 12 m/s

Adding vectors 1-dimension

We can combine vectors to make a new a vector. The new vector is called the RESULTANT vector.

Simples rule for adding in one-dimension.

If they are going in opposite directions, subtract the magnitude (value) of the vectors.

+12 m/s 7 m/s 5 m/s

=

Adding vectors 2-dimensions

There are two methods to adding vectors in 2-Dimensions.Method 1: Head-to-tail (aka. Tip-to-tail)

Adding vectors 2-dimensions

Head of blue

Tail of red

Resultant:Sum of vector red and vector blue.Connect tail of the first to the head of the second.

Note: When adding vectors head-to-tail method, order does not matter.

Make a copy of the vectors, then arrange them head to tail.

Adding vectors 2-dimensions

Method 2: Parallelogram methodThe diagonal of the parallelogram is the resultant.

Make a copy of the vectors, then arrange them into a parallelogram when the two original vectors are tail to tail.

Calculating vector sum in 2-dimensions

If the vectors are perpendicular to each other, then the sum of the two vectors can be computed by using the Pythagorean theorem.

4 m/s 3 m/s

Calculating vector sum in 2-dimensions

Using head to tail method, we can add the two vectors graphically and the resultant is the hypotenuse of the right triangle.

4 m/s

3 m/s

To find the length of the hypotenuse, use the Pythagorean Theorem.a2 + b2 = c2 where a and b are the two sides and c is the hypotenuse.

c = √(a2 + b2) = √(32 + 42)=√(9 + 16)=√(25)=5

5 m/s

Calculating vector sum in 2-dimensions

To add non-perpendicular vectors, it would require trigonometry and the use of sine and cosine to determine the horizontal and vertical components of a vector.

To bypass the use of trigonometry, use a graph paper to draw out the vectors accurately and to scale.

Vector Resolution

Any diagonal vectors (has an angle with the horizontal) can be resolved into its horizontal and vertical components.OrThink of it as being a diagonal vector is made up of two vectors, a horizontal and a vertical vector.

20 N

53o

Vector Resolution

Draw a horizontal and vertical line to form a right triangle.

20 N

53o

Vector Resolution

Impose the vector onto graph paper.

We can see that get to the head of the

vector from the tail, we move 12 spaces

to the right and 16 spaces up.

12

16

Vector Resolution

We can say that the 20 N vector at 53o with

the horizontal can be resolved into a

horizontal vector of 12 N and a vertical

vector of 16 N.

Or

We can say that the 20 N vector is made up

of a 12 N horizontal vector and a 16 N

vertical vector.12

16

Resultant vs. Resolution

Resultant - the sum of vectors. The “result” of adding or putting vectors together.

Resolution - taking apart a vector. “Resolve” or “breaking up” a vector into its components/parts.

Positive and Negative Vectors

Application

Two forces are applied to slide a crate across the floor. A 30 N force is applied in a northerly direction and a 50 N force is applied in an easterly direction. Determine the resultant of the two forces acting on the crate.

Application

1. Draw a top view free-body diagram to scale of the forces on the crate.2. Draw a resultant (net force) on the free-body diagram using either the head-to-tail method or the parallelogram method.3. Calculate the net force on the crate using the Pythagorean Theorem.

Example

30 N

50 N

Scale: 10 N

Example

Use head to tail method to determine the resultant graphically.

30 N

50 N

Scale: 10 N

30 N

Example

Use the Pythagorean Theorem to determine the magnitude of the resultant.

30 N

50 N

Scale: 10 N

30 N

Example

Use trigonometry to determine the direction of the vector. (you will not need to do this for this class)

30 N

50 N

30 N

To calculate the angle of the net force, we need to use the definition of tangent.

Tangent of an angle is the ratio of the opposite side of a right triangle to its adjacent side.