Vectors

Vectors or Scalars ?  What is a scalar?  A physical quantity with magnitude ONLY  Examples: time, temperature, mass, distance, speed  What is a vector?  A physical quantity with BOTH magnitude and direction.  Examples: weight, velocity, displacement, force

How is a vector represented?  An arrow is used to represent a vector. The length of the arrow represents the magnitude and the head of the vector represents the direction.  NOTE: a scalar is the magnitude of a vector quantity

Comparing vectors and scalars Dimension symbol vector or scalar? Time t scalar Mass m scalar Distance d scalar Displacement Δx vector Speed s scalar Velocity v vector Acceleration a vector Force F vector

Distance: A Scalar Quantity A scalar quantity: Contains magnitude only and consists of a number and a unit. (20 m, 40 mi/h, 10 gal) A B  Distance is the length of the actual path taken by an object. s = 20 m

Displacement—A Vector Quantity A vector quantity: Contains magnitude AND direction, a number, unit & angle. (12 m, 30 0 ; 8 km/h, N) A B D = 12 m, 20 o Displacement is the straight-line separation of two points in a specified direction.Displacement is the straight-line separation of two points in a specified direction. 

Distance and Displacement Net displacement: 4 m,E 6 m,W D What is the distance traveled? 10 m !! D = 2 m, W Displacement is the x or y coordinate of position. Consider a car that travels 4 m, E then 6 m, W.Displacement is the x or y coordinate of position. Consider a car that travels 4 m, E then 6 m, W. x= +4 x = +4 x= -2 x = -2

Vector Composition (vector addition)  When two or more vectors are added the directions must be considered.  Vectors may be added Graphically or Analytically.  Graphical Addition requires the use of scale drawings of vectors tip-to-tail. (Rulers and protractors are used.)  Analytical Addition is a strictly mathematical method using trigonometric functions (sin, cos, tan) to add the vectors together.

Vector addition- Graphical B A Re-draw as tip to tail by moving one of the vectors to the tip of the other. A B R R = Resultant vector which is the vector sum of A+B.The resultant always goes from the beginning (tail of first vector) to the end (tip of last vector). Given vectors A and B

 To subtract vectors, add a negative vector.  A negative vector has the same magnitude and the opposite direction. Example: A -A Note: A + (-A) = 0 So the resultant (R) is 0.

Graphical Addition of Multiple Vectors A B C Re-draw vectors tip-to-tail A B C R R = A + B + C   (direction of R from x axis) Given vectors A, B, and C Note:  represents the angle between vector B and the horizontal axis

Vector Subtraction A -B  Given : Vectors A and B, find R = A - B Re-draw tip to tail as A+(-B)  A R B Note: -B is the same magnitude but is 180 o from the original direction.

Equilibrant  The equilibrant vector is the vector that will balance the combination of vectors given.  It is always equal in magnitude and opposite in direction to the resultant vector.

Equilibrant (continued) Given vectors A and B, find the equilibrant A B Re-draw as tip to tail, find resultant, then draw equilibrant equal and opposite. A B R E

Right Triangle Trigonometry B Aadjacent opposite hypotenuse sin  = opposite hypotenuse cos  = adjacent hypotenuse tan  = opposite adjacent  And don’t forget: A 2 + B 2 = C 2 C

Vector Resolution A Ay AxAx  Given: vector A at angle  from horizontal. Resolve A into its components. (A x and A y ) x y Evaluate the triangle using sin and cos. cos  =A x /A so…A x = A cos  sin  = A y /A so… A y = A sin  Hint: Be sure your calculator is in degrees!

Vector Addition-Analytical  resolve the vectors to be added into their x- and y- components.  Add the x- components together to get a resultant vector in the x direction  Add the y- components together to get a resultant vector in the y direction  Use the pythagorean theorem to add the resultant vectors in the x- and y-components together.  Use the tan function of your resultant triangle to find the direction of the resultant. To add vectors mathematically:

Vector Addition-Analytical Example: Given: Vector A is 90 at 30 O and vector B is 50 at 125 O. Find the resultant R = A + B mathematically. 30 O 55 O A B A x = 90 cos 30 O = 77.9 A y = 90 sin 30 O = 45 B x = -50 cos 55 O = B y = 50 sin55 O = 41 Note: B x will be negative because it is acting along the -x axis. ByBy AxAx AyAy BxBx First, calculate the x and y components of each vector.

Vector Addition-Analytical (continued) Then find R: R 2 = R x 2 + R y 2 R 2 = (49.2) 2 +(86) 2, so… R = 99.1 To find direction of R:  = tan -1 ( R y / R x )  = tan -1 ( 86 / 49.2 ) = 60.2 O from x axis Find R x and R y : R x = A x + B x R y = A y + B y R x = (- 28.7) = 49.2 R y = = 86 RxRx RyRy R 

Stating the final answer  All vectors must be stated with a magnitude and direction.  Angles must be specified according to compass directions( i.e. N of E) or adjusted to be measured from the +x-axis(0°).  The calculator will always give the angle measured from the closest horizontal axis.  CCW angles are +, CW are - * * ccw = counter- clockwise cw = clockwise