Vectors

A study of motion will involve the introduction of a variety of quantities that are used to describe the physical world. Examples of such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can by divided into two categories - vectors and scalars.

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.

Recall:

vector velocity = 30 mph north

scalar speed = 30 mph

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.

velocity = 30 mph north

Scale: 1 cm = 10 mph

N

S

W E

head

tail

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 vectors are even directed northeast, yet more north than east. 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

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

0O

90O

180O

270O

Example:

Draw a 60 m/s vector rotated counterclockwise 120o.

Example:

Draw a 60 m/s vector rotated counterclockwise 120o.

0O

90O

180O

270O

10 m/s

Scale

120o

Adding vectors in 1-Dimension

If vectors in the same direction, add the magnitude of the vectors.

The sum of the vectors is called the resultant.

If the vectors are in opposite direction, subtract the magnitude of the vectors.

Adding vectors in 2-Dimensions require a bit more work.

There are two methods to adding vectors in 2-Dimensions.

Method 1: Head-to-tail method

Copy the vectors and placed them in a head to tail arrangement.

a

b

Adding vectors in 2-Dimensions require a bit more work.

There are two methods to adding vectors in 2-Dimensions.

Method 1: Head-to-tail method

Move vector a to the tail of vector b. The sum of the two vectors is the vector connecting from tail of vector a to the head of vector b.

a

b

a + b

Adding vectors in 2-Dimensions require a bit more work.

There are two methods to adding vectors in 2-Dimensions.

Method 2: Parallelogram method

Copy the vectors and connect them tail-to-tail.

a b

Adding vectors in 2-Dimensions require a bit more work.

There are two methods to adding vectors in 2-Dimensions.

Method 1: Parallelogram method

Create a parallelogram by copying another set of vectors (dotted). Connect the corner of the tails to the corner with the two heads to create the resultant.

a

b

a + b

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

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.

Vector resolution

Any diagonal vectors (has an angle with the horizontal) can be resolved into its horizontal and vertical components.

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

Vector resolution

Any diagonal vectors (has an angle with the horizontal) can be resolved into its horizontal and vertical components.

Make a vertical vector the same length as the height of the right triangle and place it tail-to-tail with the original vector.

Make a horizontal vector along the base of the right triangle. The bluevector and the red vectors are the components of the green vector.

Example:

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.

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.

10 N

Scale

30 N

50 N

Head-to-tail method

10 N

Scale

30 N

50 N

resultant

Parallelogram method

10 N

Scale

30 N

50 N

Pythagorean Theorem

10 N

Scale

a = 30 N

b = 50 NNc

c

c

c

bac

cba

31.58

3400

2500900

5030 22

22

222

c

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.

a = 30 N

b = 50 N

c

q