1 Vectors and Two-Dimensional Motion. 2 Vector Notation When handwritten, use an arrow: When handwritten, use an arrow: When printed, will be in bold.

1

Vectors and Two-Dimensional

Motion

2

Vector Notation

When handwritten, use an arrow: When printed, will be in bold print:

A When dealing with just the

magnitude of a vector in print, an italic letter will be used: A

A

3

Properties of Vectors

Equality of Two Vectors Two vectors are equal if they have

the same magnitude and the same direction

Movement of vectors in a diagram Any vector can be moved parallel to

itself without being affected

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More Properties of Vectors

Negative Vectors Two vectors are negative if they

have the same magnitude but are 180° apart (opposite directions)

A = -B

Resultant Vector The resultant vector is the sum of a

given set of vectors

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Adding Vectors

When adding vectors, their directions must be taken into account

Units must be the same Graphical Methods

Use scale drawings Algebraic Methods

More convenient

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Adding Vectors Graphically (Triangle or Polygon Method)

Choose a scale Draw the first vector with the

appropriate length and in the direction specified, with respect to a coordinate system

Draw the next vector with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of vector A and parallel to the coordinate system used for A

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Graphically Adding Vectors, cont.

Continue drawing the vectors “head-to-tail”

The resultant is drawn from the origin of A to the end of the last vector

Measure the length of R and its angle Use the scale factor to

convert length to actual magnitude

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Graphically Adding Vectors, cont.

When you have many vectors, just keep repeating the process until all are included

The resultant is still drawn from the origin of the first vector to the end of the last vector

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Alternative Graphical Method

When you have only two vectors, you may use the Parallelogram Method

All vectors, including the resultant, are drawn from a common origin The remaining sides

of the parallelogram are sketched to determine the diagonal, R

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Notes about Vector Addition

Vectors obey the Commutative Law of Addition The order in which

the vectors are added doesn’t affect the result

EXAMPLE 3.1 Taking a Trip

A graphical method for finding the resultant

displacement vector = + .

Goal Find the sum of two vectors by using a graph. Problem A car travels 20.0 km due north and then 35.0 km in a direction 60° west of north, as in the figure. Using a graph, find the magnitude and direction of a single vector that gives the net effect of the car's trip.

This vector is called the car's resultant displacement. Strategy Draw a graph and represent the displacement vectors as arrows. Graphically locate the vector resulting from the sum of the two displacement vectors. Measure its length and angle with respect to the vertical.

SOLUTION

Let represent the first displacement vector, 20.0 km north, and the second displacement vector, extending west of north. Carefully graph the two vectors, drawing a resultant vector with its base touching the base of and extending to the tip of . Measure the length of this vector, which turns out to be about 48 km. The angle β, measured with a protractor, is about 39° west of north. LEARN MORE

Remarks Notice that ordinary arithmetic doesn't work here: the correct answer of 48 km is not equal to 20.0 km + 35.0 km = 55.0 km! Question Suppose two vectors are added. The sum of the magnitudes of the two vectors is equal to the magnitude of the resultant vector whenever the two vectors are: (Select all that apply.)

at 45° to each other of equal magnitude anti-parallel (in

opposite directions) perpendicular parallel (in the same direction)

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Vector Subtraction

Special case of vector addition

If A – B, then use A+(-B)

Continue with standard vector addition procedure

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Multiplying or Dividing a Vector by a Scalar

The result of the multiplication or division is a vector

The magnitude of the vector is multiplied or divided by the scalar

If the scalar is positive, the direction of the result is the same as of the original vector

If the scalar is negative, the direction of the result is opposite that of the original vector

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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

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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,

cosAxA

sinAA y

yx AA A

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More About Components of a Vector

The previous equations are valid only if θ is measured with respect to the x-axis

The components can be positive or negative and will have the same units as the original vector

The components are the legs of the right triangle whose hypotenuse is A

May still have to find θ with respect to the positive x-axis

x

y12y

2x A

AtanandAAA

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Adding Vectors Algebraically

Choose a coordinate system and sketch the vectors

Find the x- and y-components of all the vectors

Add all the x-components This gives Rx: xx vR

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Adding Vectors Algebraically, cont.

Add all the y-components This gives Ry:

Use the Pythagorean Theorem to find the magnitude of the Resultant:

Use the inverse tangent function to find the direction of R:

yy vR

2y

2x RRR

x

y1

R

Rtan

EXAMPLE 3.2 Help Is on the Way!

Goal Find vector components, given a magnitude and direction, and vice versa. Problem (a) Find the horizontal and vertical components of the 1.00 102 m displacement of a superhero who flies from the top of a tall building along the path shown in Figure a. (b) Suppose instead the superhero leaps in the other

direction along a displacement vector to the top of a flagpole where the displacement components are given by

Bx = -25.0 m and By = 10.0 m. Find the magnitude and direction of the displacement vector.

Strategy(a) The triangle formed by the displacement and its components is shown in Figure b. Simple trigonometry gives

the components relative to the standard x-y coordinate

system: Ax = A cos θ and Ay = A sin θ. Note that θ = -30.0°,

negative because it's measured clockwise from the positive x-axis. (b) Apply the proper equations to find the magnitude and direction of the vector.

SOLUTION

(a) Find the vector components of from its magnitude and direction. Use the demonstrated equations to find the components of the displacement vector

. Ax = A cos θ = (1.00 102 m) cos (-30.0°) = +86.6 m

Ay = A sin θ = (1.00 102 m) sin (-30.0°) = -50.0 m

(b) Find the magnitude and direction of the displacement vector from its components.

Compute the magnitude of from the Pythagorean theorem.

B = √Bx2 + By

2 = √(-25.0 m)2 + (10.0 m)2 = 26.9 m

Calculate the direction of using the inverse tangent, remembering to add 180° to the answer in your calculator window, because the vector lies in the second quadrant.

θ = tan-1 ( By ) = tan-1 (

10.0 ) = -21.8° Bx -25.0

θ = 158°

LEARN MORE

Remarks In part (a), note that cos (-θ) = cos θ; however, sin (-θ) = -sin θ. The negative sign of Ay reflects the fact that displacement in the y-direction is downward. Question What other functions, if any, can be used to find the angle in part (b)? (Select all that apply.)

None of those listed.

EXAMPLE 3.3 Take a Hike

(a) Hiker's path and the resultant vector. (b) Components of the hiker's total displacement from camp.

Goal Add vectors algebraically and find the resultant vector. Problem A hiker begins a trip by first walking 25.0 km 45.0° south of east from her base camp. On the second day she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger's tower. (a) Determine the components of the hiker's displacements in the first and second days. (b) Determine the components of the hiker's total displacement for the trip. (c) Find the magnitude and direction of the displacement from base camp.

Strategy This problem is just an application of vector addition using components.

We denote the displacement vectors on the first and second days by and , respectively. Using the camp as the origin of the coordinates, we get the vectors

shown in Figure a. After finding x- and y-components for each vector, we add them "componentwise." Finally, we determine the magnitude and direction of the

resultant vector , using the Pythagorean theorem and the inverse tangent function.

SOLUTION

(a) Find the components of . Use the demonstrated equations to find the components of . Ax = A cos (-45.0°) = (25.0 km)(0.707)= 17.7 km

Ay = A sin -(45.0°) = (-25.0 km)(0.707)= -17.7 km

Find the components of . Bx = B cos 60.0° = (40.0 km)(0.500)= 20.0 km

By = B sin 60.0° = (40.0 km)(0.866)= 34.6 km

(b) Find the components of the resultant vector, = + .

To find Rx, add the x-components of and .

Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km

To find Ry, add the y-components of and .

Ry = Ay + By = -17.7 km + 34.6 km = 16.9 km

(c) Find the magnitude and direction of .

Use the Pythagorean theorem to get the magnitude.

R = √Rx2 + Ry

2 = √ (37.7 km2 + (16.9 km)2 = 41.3 km

Calculate the direction of using the inverse tangent function.

θ = tan-1 ( 16.9 km ) = 24.1°

LEARN MORE

Remarks Figure b shows a sketch of the components of and their directions in space. The magnitude and direction of the resultant can also be determined from such a sketch. Question A second hiker follows the same path the first day, but then walks 15.0 km east on the second day before turning and reaching the ranger's tower. How does the second hiker's resultant displacement vector compare with that of the first hiker? List all aspects that apply. (Select all that apply.)

The two displacements have the same direction. The second hiker's displacement

has the same magnitude as the first. The second hiker's displacement is greater in magnitude.

The two displacements have different directions. The second hiker's displacement has a smaller magnitude than the first.