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Vectors

Objectives:

• Define the concept of a vector

• Learn how to perform basic vector operations using graphical and numerical methods

• Learn how to use vector algebra to solve simple problems

Vector

A vector is a quantity that has:

• a magnitude

• a direction

(e.g. change in position)

A scalar quantity has magnitude only (e.g. time)

V

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Numerical RepresentationMethods of expressing a vector (V) numerically:• its magnitude (V) and direction (θ) with respect

to a reference axis• its components (Vx, Vy) along each reference

axis

V

(0,0) xVx

Vy

θ

y

V

Vector Composition

resultant : vector resulting from the composition

• Process of determining a single vector from two or more vectors by vector addition

• Performed graphically using tip-to-tail method

V1

V2

copy ofV2

V1 + V2

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Vector Resolution

• Process of replacing a single vector with two perpendicular vectors whose composition equals the original vector

V

V1

V2V

V1

V2

Another Resolution of V

Resolution into Components

• Trigonometry can be used to numerically resolve a vector into its x- and y-components

V

(0,0) xθ

y

V

cos θ = Vx

V

sin θ = Vy

V

Vy = V * sin θ

Vx = V * cos θVy

Vx

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Composition of Components

• A vector can be numerically composed from its components using geometry and trigonometry

V

(0,0) xθ

y

V Vy

Vx

θ = atan Vy

Vx

V = Vx2 + Vy

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Composition of 1-Dimensional Vectors• Vectors pointing in same direction:

V1 + V2

V1 V2

• Vectors pointing in opposite direction:V1V2

• magnitudes sum,• direction remains same

V1 + V2 • magnitudes subtract:(larger – smaller),

• direction is that oflarger vector

V1

V2V1 + V2

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Numerical Vector Composition

1. Draw x- and y-axes2. Resolve each vector into x and y components3. x component of resultant = add each

component pointing in +x direction and subtract each component pointing in –x direction.

4. y component of resultant = add each component pointing in +y direction and subtract each component pointing in –y direction.

5. Draw the x and y components of the resultant6. Compose the resultant from its components

V1

V2

θ2x

y

θ1

V2y = V2 sin θ2

V2x = V2 cos θ2

V1y = V1 sin θ1

V1x = V1 cos θ1

VRx = V1x – V2x

VRy = V1y + V2y

VRx

VRy

VR

θR = atan (VRy / VRx )

VR = VRx2 + VRy

2

θR

V2

Example

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Alternate Method of Composition

1. Draw vectors “tip-to-tail”2. Draw resultant vector to form a triangle3. Draw x- and y-axes at tail of first vector4. Determine the angle between the first and

second vector in the triangle.5. Use Law of Cosines to determine the

magnitude of the resultant. 6. Use Law of Sines to determine the angle

between the first vector and the resultant7. Compute direction of the resultant from

identified angles

V1

V2

x

y

θ1

VR

V2

Example

αβ

sin αVR

sin βV2

=

β = asinV2 sin α

VR

θR

θR = θ1 + β

VR = V12 + V2

2 – 2V1V2 cos α

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Vector-Scalar MultiplicationIf a vector V is multiplied by a scalar n:• If n > 0:

– magnitude of resultant = n * V– direction of resultant = direction of V

• If n < 0:– magnitude of resultant = (–n) * V– direction of resultant = opposite direction of V

V3 * V

-1 * V

θθ

Vector Subtraction

• Subtraction of a vector performed by adding (–1) times the vector

• Can be performed graphically or numerically

V1

V2

V1 – V2-1 * V2

-1 * V2

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Subtraction as a Change

• Subtraction can be pictured as the difference or change between two vectors that originate from the same point

V1

V2

V2 – V1

x

y V1 + (V2 – V1) = V2

Graphical Solution Using Vectors

1. Establish a scaling factor for the graph (e.g. 1cm = 10 m/s)

2. Carefully draw vectors with the correct length (based on the scaling factor) and direction

3. Use graphical methods of composition, resolution, scalar multiplication, and/or subtraction to find desired resultant

4. Carefully measure the length and direction of the resultant.

5. Use scaling factor to convert measured length to magnitude

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Example Problem #1

Two volleyball players simultaneously contact the ball above the net.

Player #1 hits the ball from the left with a force of 300 N (67 lb), angled 45° below the horizontal.

Player #2 hits the ball from the right with a force of 250 N (56 lb), angled 20° below the horizontal.

What is the magnitude and direction of the net force applied to the ball by the 2 players?

Numerical Solutions Using Vectors

1. Sketch the vectors on a diagram of the problem

2. Choose and diagram the coordinate axes, based on:

• axes used in the problem statement• axes that are physically meaningful

3. Establish and label known magnitudes and angles or x- and y-components

4. Use numerical methods of composition, resolution, scalar multiplication, and/or subtraction to find desired solution

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Graphical vs. Numerical Method

• Graphical Method– Simple– Must be done by hand– Gives approximate result

• Numerical Method– Requires complex calculations– Gives accurate result– Can be performed by computer– Can perform analyses in 3 dimensions

Example Problem #2

A golfer is teeing off from the center of the fairway for a hole that is located 300 yards away and 30° to the right of center.

The golfer’s tee shot goes 210 yards and 15° to the left of center of the fairway.

To reach the hole on the second shot, how far and in what direction must the golfer hit the ball?