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Understanding Vectors: Definition, Examples & Operations

Learn the fundamentals of vectors - physical quantities with magnitude and direction. Explore vector representation, vector addition, subtraction, and resolution, with examples and analytical methods.

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Understanding Vectors: Definition, Examples & Operations

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  1. Vectors

  2. 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

  3. 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

  4. 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

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

  6. B D = 12 m, 20o A q Displacement—A Vector Quantity • Displacement is the straight-line separation of two points in a specified direction. A vector quantity: Contains magnitude AND direction, a number,unit & angle. (12 m, 300; 8 km/h, N)

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

  8. 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.

  9. Vector addition- Graphical Given vectors A and B Re-draw as tip to tail by moving one of the vectors to the tip of the other. R B B A A 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).

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

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

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

  13. 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.

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

  15. Right Triangle Trigonometry sin  = opposite hypotenusecos  = adjacent hypotenusetan  = opposite adjacent hypotenuse opposite C B  A adjacent And don’t forget: A2 + B2 = C2

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

  17. Vector Addition-Analytical To add vectors mathematically: • 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.

  18. Vector Addition-Analytical Given: Vector A is 90 at 30O and vector B is 50 at 125O. Find the resultant R = A + B mathematically. Example: B Ax= 90 cos 30O = 77.9 Ay= 90 sin 30O = 45 Bx = -50 cos 55O = -28.7 By = 50 sin55O = 41 Note: Bx will be negative because it is acting along the -x axis. A By Ay 55O 30O Bx Ax First, calculate the x and y components of each vector.

  19. Vector Addition-Analytical(continued) Find Rx and Ry: Rx = Ax + Bx Ry = Ay + By Rx = 77.9 + (- 28.7) = 49.2 Ry = 41 + 45 = 86 R Ry  Then find R: R2 = Rx2 + Ry2 R2 = (49.2)2+(86)2 , so… R = 99.1 To find direction of R:  = tan-1 ( Ry / Rx )  = tan-1 ( 86 / 49.2 ) = 60.2O from x axis Rx

  20. 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

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