Vectors Chapter 3 Sections 1 and 2 Vectors

Vectors Chapter 3, Sections 1 and 2

Vectors and Scalars • Measured quantities can be of two types • Scalar quantities: only require magnitude (and proper unit) for description. Examples: distance, speed, mass, temperature, time • Vector quantities: require magnitude (with unit) and direction for complete description. Examples: displacement, velocity, acceleration, force, momentum

Vector Addition • When 2 or more vectors act on an object, the total effect is the vector sum • Special math operations must be used with vectors • Vector sum called the resultant • Sum can be found using graphic methods (drawing to scale) or mathematical methods

Adding Vectors Graphically • Choose a suitable scale for the drawing • Use a ruler to draw scaled magnitude and a protractor for the direction • Vectors can be moved in a diagram as long as their length and direction are not changed • Vectors can be added in any order without changing the result

Adding Vectors Graphically • Use head-to-tail method for series of sequential vectors where each successive vector begins where the preceding vector ended • Also works for two vectors acting simultaneously at the same point, although drawing doesn’t match the physical situation

Adding Vectors Graphically • Resultant is a vector drawn from point of origin to tip of last vector • Magnitude of resultant can be found by measuring and converting the measurement using the scale of the drawing • Direction is found by measuring angle with a protractor

Adding Vectors Graphically • Graphical method gives approximate values depending on drawing accuracy One dimensional graphical addition of vectors

Subtracting Vectors • To subtract a vector, add a negative vector, one having same magnitude but opposite direction a b b a +b a -b a a-b

Parallelogram Method • Vectors are drawn from a common origin • Complete parallelogram by drawing opposite sides parallel to vectors • Resultant is the diagonal of the parallelogram

Pros and Cons for Graphical Methods • For simultaneous vectors like forces, parallelogram method gives a better picture of actual situation • More difficult to draw accurately • Better for sketches, not for measured drawings • Head-to-tail method better for measuring

Parallelogram Method • a+b=r a r b

Adding Vectors Mathematically • Exact values for vector sums using trig functions (tan mostly) and Pythagorean theorem • Set up vectors on x-y coordinate system • If vectors act at right angles, Pythagorean theorem gives resultant magnitude • Direction can be found with tan-1 function

Resolving Vectors Into Components • A vector acting at an angle to the coordinate axes can be resolved into x and y components that would add together to equal the original vector • The x-component = original magnitude times the cos of the angle measured from the x-axis • The y-component = original magnitude times the sin of the angle measured from the x-axis

Vector Components vx= (50 m/s)(cos 60 o) vy= (50 m/s)(sin 60 o)

Adding Non-perpendicular Vectors • Resolve each vector into x and y components • Add the x-components together and add the y-components together • Use Pythagorean theorem and tan-1 function to find magnitude and direction of resultant

Adding Non-perpendicular Vectors y y x x

Adding Non-perpendicular Vectors Rx = 11. 3 + 12. 5 = 23. 8 Ry= 4. 1 + 21. 7 = 25. 8 @

Alternate Method for Adding Nonperpendicular Vectors • Consider a vector triangle with angles A, B, and C with opposites sides labeled a, b, and c A c B b C a

Alternate Method for Adding Nonperpendicular Vectors • Cosine law can be used to find magnitude of vector c if magnitudes and directions of a and b are known • Angle between a and b can be found using simple geometry • Sine law can be used to find direction of vector c

Cosine Law • Useful if two sides (a and b) and the angle between them (C) are known: • Similar to Pythagorean Theorem with a correction factor for lack of right angle

Sine Law • Useful when one angle and its opposite side are known along with one other side or angle