1.Total degrees in a triangle: 2.Three angles of the triangle below: 3.Three sides of the triangle below: 4.Pythagorean Theorem: x 2 + y 2 = r A B C x, y, and r y x r HYPOTENUSE A, B, and C

Trigonometric functions are ratios of the lengths of the segments that make up angles. Q y x r sin Q = = opp. y hyp. r cos Q = = adj. x hyp. r tan Q = = opp. y adj. x

sin A = opposite hypotenuse cos A = adjacent hypotenuse tan A = opposite adjacent sin A = 1212 cos A = tan A = √ A B C 1 √3 For <A below, calculate Sine, Cosine, and Tangent:

a c A B C b Law of Cosines: c 2 = a 2 + b 2 – 2ab cos C Law of Sines: sin A sin B sin C a b c =

1.Scalar – a variable whose value is expressed only as a magnitude or quantity Height, pressure, speed, density, etc. 2.Vector – a variable whose value is expressed both as a magnitude and direction Displacement, force, velocity, momentum, etc. 3. Tensor – a variable whose values are collections of vectors, such as stress on a material, the curvature of space-time (General Theory of Relativity), gyroscopic motion, etc.

Properties of Vectors 1.Magnitude Length implies magnitude of vector 2.Direction Arrow implies direction of vector 3.Act along the line of their direction 4.No fixed origin Can be located anywhere in space

Magnitude, Direction Vectors - Description 45 o 40 lbs F = 40 lbs 45 o F = o magnitude direction Hat signifies vector quantity Bold type and an underline F also identify vectors

1.We can multiply any vector by a whole number. 2.Original direction is maintained, new magnitude. Vectors – Scalar Multiplication 2 ½

1.We can add two or more vectors together. 2.2 methods: 1.Graphical Addition/subtraction – redraw vectors head-to- tail, then draw the resultant vector. (head-to-tail order does not matter) Vectors – Addition

Vectors – Rectangular Components y x F FxFx FyFy 1.It is often useful to break a vector into horizontal and vertical components (rectangular components). 2.Consider the Force vector below. 3.Plot this vector on x-y axis. 4.Project the vector onto x and y axes.

Vectors – Rectangular Components y x F FxFx FyFy This means: vector F = vector F x + vector F y Remember the addition of vectors:

Vectors – Rectangular Components y x F FxFx FyFy F x = F x i Vector F x = Magnitude F x times vector i Vector F y = Magnitude F y times vector j F y = F y j F = F x i + F y j i denotes vector in x direction j denotes vector in y direction Unit vector

Vectors – Rectangular Components y x F FxFx FyFy Each grid space represents 1 lb force. What is F x ? F x = (4 lbs)i What is F y ? F y = (3 lbs)j What is F? F = (4 lbs)i + (3 lbs)j

Vectors – Rectangular Components If vector V = a i + b j + c k then the magnitude of vector V |V| =

Vectors – Rectangular Components F FxFx FyFy cos Q = F x / F F x = F cos Q i sin Q = F y / F F y = F sin Q j What is the relationship between Q, sin Q, and cos Q? Q

Vectors – Rectangular Components y x F F x + F y + When are F x and F y Positive/Negative? F F x - F y + F F F x - F y - F x + F y -

Vectors – Rectangular Components Complete the following chart in your notebook: I II III IV

1.Vectors can be completely represented in two ways: 1.Graphically 2.Sum of vectors in any three independent directions 2.Vectors can also be added/subtracted in either of those ways: 1. 2.F 1 = ai + bj + ck; F 2 = si + tj + uk F 1 + F 2 = (a + s)i + (b + t)j + (c + u)k Vectors

A third way to add, subtract, and otherwise decompose vectors: Use the law of sines or the law of cosines to find R. Vectors F1F1 F2F2 R 45 o 105 o 30 o

Brief note about subtraction 1.If F = ai + bj + ck, then – F = – ai – bj – ck 2.Also, if F = Then, – F = Vectors

Resultant Forces Resultant forces are the overall combination of all forces acting on a body. 1) find sum of forces in x-direction 2) find sum of forces in y-direction 3) find sum of forces in z-direction 3) Write as single vector in rectangular components R = S F x i + S F y j + S F z k

Resultant Forces – Example 1 A satellite flies without friction in space. Earth’s gravity pulls downward on the satellite with a force of 200 N. Stray space junk hits the satellite with a force of 1000 N at 60 o to the horizontal. What is the resultant force acting on the satellite? 1.Sketch and label free-body diagram (all external and reactive forces acting on the body) 2.Decompose all vectors into rectangular components (x, y, z) 3.Add vectors

Example 2 A stop light is held by two cables as shown. If the stop light weighs 120 N, what are the tensions in the two cables?