Physical Quantities are of 2 types… A scalar is only a magnitude (length)

(Example: Temperature, time, mass)

A vector has magnitude and direction (Example: displacement = 10 m East, Velocity= 50 mph west)

A

A vector will be symbolized by the “letter” with an arrow over it. The arrow indicates direction.

Vectors are equal if they have the same units, magnitude, and direction.

A vector can be moved anywhere parallel to itself.

Adding Vectors (attach) 2 displacements carried out in succession result in a

net displacement, the vector sum of the individual displacements

To add vectors they must have the same units.

Head -to -tail method put them head to tail and connect them so you end up with a triangle.

Resultant Vector

The resultant vector is the sum of a given set of vectors

C = A + B

Subtracting Vectors head to tail- subtract by

putting vector in the opposite direction

If you change the sign of a vector it is not the same vector. It is a new vector.

A – B does not equal

B - A

Pythagorean Theorem

Components of a Vector

A component is a part

It is useful to use rectangular components

These are the projections of the vector along the x- and y-axes

Components of a Vector, cont. The x-component of a vector is the projection along

the x-axis

The y-component of a vector is the projection along the y-axis

Then,

cosA A x

sinyA A

x y A A A

Useful Formulas..

2 2 1tany

x y

x

AA A A and

A

The Pythagorean Theorem can only be used

with right triangles!

Example 1 Find the magnitude of the sum of a 15 km

displacement and a 25 km displacement when the angle between them is 900 and when the angle between them is 1350.

Example 2 (a) Find the horizontal and vertical

components of the 100m displacement of a superhero who flies from the top of a tall building at an angle of 30.00

(b) (b) Suppose instead the superhero leaps in the other direction along a displacement vector B to the top of a flagpole where the displacement components are given Bx = -25m and BY=10.0m. Find the magnitude and direction of the displacement vector.

Example 3 A GPS receiver indicates that your home is 15.0 km and

400 north of west, but the only path through the woods leads directly north. If you follow the path 5.0 km before it opens into a field, how far, and in what direction, would you have to walk to reach your home?

R= 12.39

Ө= 158’

Resolving a Vector Into Components

+x

+y

A

Ax

Ay

The horizontal, or

x-component, of A is

found by Ax = A cos .

The vertical, or

y-component, of A is found by Ay = A sin .

By the Pythagorean Theorem, Ax2 + Ay

2 = A2.

Every vector can be resolved using these

formulas, such that A is the magnitude of A, and

is the angle the vector makes with the x-axis.

Each component must have the proper “sign”

according to the quadrant the vector terminates in.

Analytical Method of Vector Addition

1. Find the x- and y-components of each vector. Ax = A cos = Ay = A sin =

Bx = B cos = By = B sin = Cx = C cos = Cy = C sin =

2. Sum the x-components.

This is the x-component of the resultant.

Rx =

3. Sum the y-components.

This is the y-component of the resultant.

Ry =

4. Use the Pythagorean Theorem to find the

magnitude of the resultant vector.

Rx2 + Ry

2 = R2

A roller coaster moves 215 ft horizontally and then rises 130 ft at an angle of 35.00 above the horizontally. Next, it travels 125 ft at an angle of 50.00 below the horizontal. Find the roller coaster’s displacement from its starting point to the end of this movement.

You try this one on your own… Add the following vectors via the component method:

Vector A = 4m South, Vector B= 7.3 m Northwest

Find R and Ө

A person walks 25.0° north of east for 2.80 km. How far would the person walk due north and due east to arrive at the same location?

1.183 km North

2.53 km East

A person walks East for 255m and then 60° North of East for 100m. Find the magnitude and direction of the resultant displacement.

A vector has an x-componet of -25.0 units and a y-componet of 45.0 units. Find the magnitude and direction of the vector.

A quarter back takes the ball from the line of scrimmage, runs backwards for 15.0 yards, then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 60.0 yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football’s resultant displacement?

A Novice Golfer on the green takes three strokes to sink the ball. The successive displacement of the ball are 5.00 m to the north, 2.00 m 550 north of east, and 1.00 m at 300 west of south. Starting at the same initial point, an expert golfer could make the hole in what singe displacement?

3 Dimensional Vectors

Expressed in terms of unit vectors

Example Consider the following three vectors

A = 2i + 2j – k

B = i - 3j + 3k

C = -i + 2j + 2k

What is the resultant D = A + B + C of these three vectors? What is the magnitude of the resultant?

Vector Multiplication DOT PRODUCT scalar product

A ∙ B

A ∙ B = AB cosφ The product of the 2 vectors and the cosine of the angle between them

If φ < 90° the number will be positive

If φ > 90° the number will be negative

If φ = 90° the number will be zero

A ∙ B = (Axi + Ayj) (Bxi + Byj) = AxBx i ∙ i + AxBy i ∙ j + AyBx j ∙ i + AyBy j ∙ j i.i = j.j = k.k = 1 and i.j = j.i = i.k = k.i = j.k = k.j = 0 A ∙ B = AxBx i ∙ i + AyBy j ∙ j With 3 dimension: A ∙ B = AxBx i ∙ i + AyBy j ∙ j + AzBz k ∙ k

The displacement from Miami to the hurricane is B = (173km)i + (-100km)j; the displacement from the initial position of the reconnaissance aircraft to the hurricane is C = (173 km)i + (-200km) j. Us the dot product to determine the angle between these two vectors.

CROSS PRODUCT vector product

A x B The product of the 2 vectors and the sine of the angle between them

A x B is not the same as B x A

… the direction is opposite

Right Hand Rule

i x i = j x j = k x k = 0

i x j = k j x k = I k x i = j

A x B = (AyBz - AzBy ) i + (AzBx - AxBz ) j + (AxBy - AyBx ) k

Two vectors in component forms are written as :

In evaluating the product, we make use of the fact that multiplication of the same unit vectors gives the value of 0, while multiplication of two different unit vectors result in remaining vector with appropriate sign. Finally, the vector product evaluates to vector terms :

Example

Review Problems 91-92 (pages 66-68)