StarterIf the height is 10m and the angle is 30 degrees, how long is the shadow?

h/s = tanq, so s =h/tanq = 10/tan30 = 17.3m

Vectors and Vector Addition

1. Characteristics of Vectors2. Multiplying a vector by a scalar3. Adding Vectors Graphically4. Adding Vectors using

Components

What is a vector?

A vector is a mathematical quantity with two characteristics:

1. Magnitude or Length

2. Direction ( usually an angle)

Vectors vs. Scalars

A vector has a magnitude and direction.

Examples: velocity, acceleration, force,

torque, etc.

Vectors vs. Scalars

A scalar is just a number.

Examples: mass, volume, time, temperature, etc.

A vector is represented as a ray,or an arrow.

V

The initial end or tail

The terminal end or head

Picture of a Vector Named A

Magnitude of A

A = 10

Direction of A

q = 30 degrees

The Polar Angle for a Vector

Start at the positive x-axisand rotate counter-clockwise until you reach the vector.

That’s how you find the polar angle.

Two vectors A and B are equal if they have the same magnitude and direction.

A B

This property allows us to move vectors around on our paper/blackboard without changing their properties.

A = -B says that vectors A and B are anti-parallel. They have same size but the opposite direction.

A

B

A = -B also impliesB = -A

Graphical Addition of Vectors( Head –to Tail Addition )

To find C = A + B :1st Put the tail of B on the head of A.

2nd Draw the sum vector with its tail on the tail of A, and its head on the head of B.

Example: If C = A+B, draw C.

Here’s Vector C

Graphical Addition of Vectors( Head –to Tail Addition )

To find C = A - B :1st Put the tail of -B on the head of A.

2nd Draw the sum vector with its tail on the tail of A, and its head on the head of -B.

Example: If C = A-B, draw C.

Here’s Vector C = A - B

Addition of Many Vectors

A

BC D

AB

C

D

R

R = A + B + C + D

Add A,B,C, and D

Multiplication of a Vector by a Number.

A

2A-3A

Vector Addition by Components

(Do the math)

A vector A in the x-y plane can be represented by its perpendicular components called Ax and Ay.

x

y

A

AX

AY

Components AX and AY

can be positive, negative,or zero. The quadrantthat vector A lies indictates the sign of thecomponents.Components are scalars.

When the magnitude of vector A is given and its direction

specified then its componentscan be computed easily

x

y

A

AX

AYAX = Acosq

AY = Asinq

You must use the polar angle in these formulas.

Example: Find the x and y components of the vector shown ifA = 10 and q = 225 degrees.

AX = Acos = q 10 cos(225) = -7.07

Ay = Asin = q 10 sin(225) = -7.07

A = (-7.07, -7.07)

The magnitude and polar angle vector can be found by knowing its components

= tan-1(AY/AX) + C

A =

Example: Find A, and q if A = ( -7.07, -7.07)

== 10

= tan-1(AY/AX) + C = tan-1(-7.07/-7.07) + 180 = 225 degrees

Example: Find A, and q if A = ( 5.00, -4.00)

== 6.40

= tan-1(AY/AX) + C = tan-1(-4.00/5.00) + 360 = 321 degrees

Ax = Acosq

Ay = Asinq

If you know A and , q you can get Ax and Ay with:

If you know Ax and Ay

you can get A and q with:

A vector can be represented by its magnitude and angle, or its x and y components. You can go back and forth’from each representation with these formulas:

Adding Vectors by Components

If R = A + B

Then Rx = Ax + Bx

and Ry = Ay + By

So to add vectors, find their components and add the like components.

Example A = ( 3.00,2.00) and B = ( 0, 4.00)If R = A + B find the magnitude and direction of R.

Solution: R = A + B = ( 3.00,2.00) + ( 0, 4.00),

so R = ( 3.00, 6.00)

Then R = ( 32 + 62)1/2 = 6.70

q = tan-1( 6/3) = 63.4o

ExampleIf R = A + B find the magnitude and direction of R.

1st: Find the components of A and B.

Ax = 10cos 30 = 8.66 Ay = 10 sin30 = 5.00 Bx = 8cos 135 = -5.66 By = 8sin 135 = 5.66

2nd: Get Rx and Ry

Rx = Ax + Bx = 8.66 -5.66 = 3.00 Ry = Ay + By = 5.00 + 5.66 = 10.73rd: Get R and : q R = ( 32 + 10.72)1/2 = 11.1

q = tan-1 ( 10.7/3.00) = 74.3o

Ax = Acosq

Ay = Asinq

If you know A and , q you can get Ax and Ay with:

If you know Ax and Ay

you can get A and q with:

Summary

If R = A + B Rx = Ax + Bx

Ry = Ay + By

Unit Vectors

= ( 1, 0, 0) = ( 0, 1, 0 ) = ( 0, 0, 1)

Examples: A = (3,-2,5) = 3i - 2j + 5k

B = (3,0,5) = 3i + 5k

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The Scalar or Dot Product

Example: A = 3i +4j +5k B = 4i + 2j– 3k

A.B = 12 + 8 – 15 = 5

The Cross Product

A x B = i( Ay Bz - Az By ) - j( Ax Bz - Az Bx ) + k( Ax By - Ay Bx )

Example

A = (3,0,2) B = (1,1,0)

A X B = = i (0 -2) -j(0-2)+k(3-0)

= -2i +2j +3k = (-2,2,3)

Integrals of Vectors

dx + j

Example: If A = 3xi +5x2j , then find

+ j

Derivatives of Vectors

dA/dx = d (Ax, Ay, Az)/dx= (dAx/dx, dAy/dy, dAz/dz)

Example: If A = 3ti - 5t2j , then find dA/dt.

dA/dt = 3i -10tj

EXIT

If A = 6cos(3t)i +5cos(10t)j , then find