Properties of Scalars and Vectors
Vectors A vector contains two pieces of information: specific numerical value specific direction drawn as arrows on diagrams
Scalars can be described completely by just one numerical piece of information some are only positive quantities
Vectors Notation: Vector: A Length of vector: A or | A | A ≡ | A |
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
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
Equal Vectors have the same magnitude and the same direction A A B B A = B
A vector can be transported as long as its magnitude and direction remain unchanged C
Displacement Vector C represents the displacement A A B B C C
Operations with Vectors: Geometric Techniques
Adding Vectors If we begin with vector V... V 2V -V/2 -V
Adding Vectors If the vectors are unequal... V1V1 V2V2 R is called the resultant. R
Adding Vectors You can add more than two vectors... V1V1 V2V2 V3V3 R
Vector Subtraction The vector expressions A – B and A +(-B) are equivalent.
Vector Subtraction Graphically find A – B. A B A -B A – B
Operations with Vectors: Mathematical Techniques
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.
Similar Triangles
Right Triangle a triangle containing one right angle
Right Triangle a triangle containing one right angle the acute angles will always add up to 90°
Right Triangle a triangle containing one right angle the hypotenuse is the side opposite the right angle it is usually labeled “c”
The Pythagorean Theorem: Important Facts to Know a² + b² = c²
The sine ratio (opp/hyp): Important Facts to Know sin α = c a sin β = c b
The cosine ratio (adj/hyp): Important Facts to Know cos α = c b cos β = c a
The tangent ratio (opp/adj): Important Facts to Know tan α = b a tan β = a b
Question What is the measure of α? tan α = 8 7 tan α = α = tan -1 (7/8) α 41.2°
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
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!
Example Why do we use 2 SDs? N NxNx NyNy ° N x = 31 cos 63° = 14 units N y = 31 sin 63° = 28 units
Example Why are both components positive? N NxNx NyNy ° N x = 14 units N y = 28 units
Vector Components It is important to indicate the direction of each component. Down (y) and Left (x) are usually negative (Ex. 4-5).
Vector Components It is important to indicate the direction of each component. Sometimes compass directions are used (Ex. 4-6).
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.
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!
Vector Components Two (or more) vectors can be added by adding their components! (2) These are the components of the resultant vector
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