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Properties of Scalars and Vectors

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Vectors A vector contains two pieces of information: specific numerical value specific direction drawn as arrows on diagrams

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Scalars can be described completely by just one numerical piece of information some are only positive quantities

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Vectors Notation: Vector: A Length of vector: A or | A | A ≡ | A |

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Vector Conventions In the Cartesian plane, the reference direction is the positive x-axis Positive angles are measured counter- clockwise Negative angles are measured clockwise

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Vector Conventions Map directions are always referenced to geographic north, at the top of maps Angles referenced to true north are indicated by a capital “T” in place of the degree symbol Three digits are used

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Equal Vectors have the same magnitude and the same direction A A B B A = B

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A vector can be transported as long as its magnitude and direction remain unchanged C

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Displacement Vector C represents the displacement A A B B C C

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Operations with Vectors: Geometric Techniques

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Adding Vectors If we begin with vector V... V 2V -V/2 -V

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Adding Vectors If the vectors are unequal... V1V1 V2V2 R is called the resultant. R

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Adding Vectors You can add more than two vectors... V1V1 V2V2 V3V3 R

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Vector Subtraction The vector expressions A – B and A +(-B) are equivalent.

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Vector Subtraction Graphically find A – B. A B A -B A – B

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Operations with Vectors: Mathematical Techniques

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Similar Triangles Two triangles are similar when the three angles of one triangle have the same measures as the corresponding angles of the other triangle.

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Similar Triangles

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Right Triangle a triangle containing one right angle

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Right Triangle a triangle containing one right angle the acute angles will always add up to 90°

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Right Triangle a triangle containing one right angle the hypotenuse is the side opposite the right angle it is usually labeled “c”

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The Pythagorean Theorem: Important Facts to Know a² + b² = c²

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The sine ratio (opp/hyp): Important Facts to Know sin α = c a sin β = c b

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The cosine ratio (adj/hyp): Important Facts to Know cos α = c b cos β = c a

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The tangent ratio (opp/adj): Important Facts to Know tan α = b a tan β = a b

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Question What is the measure of α? tan α = 8 7 tan α = 0.875 α = tan -1 (7/8) α 41.2°

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Vector Components Every vector has two vector components which are perpendicular to each other. The horizontal component is given a subscript of x: V x The vertical component is given a subscript of y: V y

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Vector Components If you know the reference angle α for the vector, its components are found by: | V x | = V cos α | V y | = V sin α Assign the correct signs!

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Example Why do we use 2 SDs? N NxNx NyNy 31 63 ° N x = 31 cos 63° = 14 units N y = 31 sin 63° = 28 units

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Example Why are both components positive? N NxNx NyNy 31 63 ° N x = 14 units N y = 28 units

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Vector Components It is important to indicate the direction of each component. Down (y) and Left (x) are usually negative (Ex. 4-5).

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Vector Components It is important to indicate the direction of each component. Sometimes compass directions are used (Ex. 4-6).

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Vector Components In three dimensions, there are x-, y-, and z-components. By convention, the z-axis is vertical; the others are in the horizontal plane. In three dimensions, there are x-, y-, and z-components. By convention, the z-axis is vertical; the others are in the horizontal plane. In three dimensions, there are x-, y-, and z-components. By convention, the z-axis is vertical; the others are in the horizontal plane.

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Vector Components Two (or more) vectors can be added by adding their components! (1) Find the x- and y- components of each vector and add them Two (or more) vectors can be added by adding their components! (1) Find the x- and y- components of each vector and add them Two (or more) vectors can be added by adding their components!

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Vector Components Two (or more) vectors can be added by adding their components! (2) These are the components of the resultant vector

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Vector Components Two (or more) vectors can be added by adding their components! (3) The angle of the resultant vector can also be found with this information

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