Vectors Vector vs Scalar Quantities and Examples Vector

Vectors • Vector vs Scalar Quantities and Examples • Vector Addition – Graphical and Analytical Methods

What is a Scalar Quantity? • A scalar quantity has magnitude (amount, includes a number and unit) only. • Some examples of scalar quantities are: distance (5 m) speed (20 m/s) mass (3 kg) time (4 sec) volume (30 ml)

What is a vector quantity? • A vector quantity requires magnitude and direction to completely describe it. • An arrow or ray is used to represent a vector. The length of the arrow represents the magnitude and the arrow points in the direction of the vector. • Some examples of vectors are: displacement (5 m, North) velocity (20 m/s @ 60º) acceleration (20 m/s 2, South)

Drawing a Vector • Choose an appropriate scale for the vector. Make sure the scale doesn’t make the vector less than 2 cm long. • Using graph paper, mark your origin (starting point). If not on graph paper, make an origin like this: • Measure the angle of the vector from east (or 0° or positive x-axis) using a protractor and mark it. Draw a light line from the origin to that mark. • If the angle is beyond 180°, you must turn your protractor upside down.

Drawing a vector con’t • Starting at the origin, darken the line for the correct length according to the scale and include an arrow tip to indicate the direction. • Example: Draw 40 km @ 150° Scale: 1 cm = 10 cm 4 cm 150°

Distance vs Displacement • Distance – length, “how far”, scalar • Displacement – length and direction, “how far” in a given direction, vector, final position – initial position 3 m, E • Ex: 2 m, N +3 m, E + 2 m, S Total distance = 2 m + 3 m + 2 m = 7 m Displacement = 3 m, east 2 m, N 3 m, E displacement 2 m, S

Vector Addition • When two or more vectors are added, the directions must be considered. • Vectors may be added Graphically or Analytically. • Graphical Vector Addition requires the use of rulers and protractors to make scale drawings of vectors tip-to-tail. (Less accurate) • Analytical Vector Addition is a mathematical method using trigonometric functions (sin, cos, tan) and Pythagorean theorem.

Resultant • The resultant is the sum of the vectors being added. • The resultant vector is drawn from the tail of the first vector (origin or starting point) to the tip of the last vector (end or finish). • The angle for the resultant is measured using a protractor at the origin or starting point. If possible measure from east (0° or positive x-axis) and that is the resultant angle, since angles are assumed to be measured from east counter-clockwise.

Resultant – con’t • If not possible, measure the acute angle from N (90°), W (180°), S (270°), or (360°). • Adjust it by adding directions such as N of W. • Or by adding (or subtracting) the acute angle to (from) 90°, 180°, 270°, or 360° depending on whether the acute angle is before(subtract) or after (add) 90°, 180°, 270°, or 360°. • Ex: The angle below would be 30° S of W or 210° (180° + 30° since the 30° is past 180°).

Graphical Addition of Several Vectors. Given vectors A, B, and C • Draw each of the vectors tip-to-tail to scale. • Draw the resultant from the tail of first vector to tip of last vector (start to finish) B C R R=A+B+C A (direction of R)

Graphical Addition Resultant R=A+B+C Given vectors A, B, &C R A B C (direction of R) • To get the magnitude of the resultant, measure the length of R and multiply by the scale factor. • To get the direction of the resultant, measure the angle from 0° (or E or +x) to the resultant vector. If more than 180°, see the previous resultant slide.

1 D Vector Addition - Analytically • When adding vectors in the same direction, add their magnitudes and the resultant direction is the same. Ex: 5 m, E + 10 m, E = 15 m, E • When adding vectors in the opposite direction, subtract their magnitudes and the resultant is in the same direction as larger magnitude. Ex: 5 m, E + 10 m, W = 5 m, w

1 D Vector Addition - Analytically • To simplify a vector addition of several vectors, add all the vectors going the same way first and then subtract the ones going in the opposite direction. • Ex: 5 m, E + 10 m, W + 3 m, W + 2 m, E (add the easts and add the wests) = 7 m, E + 13 m, W (subtract the opposite directions) =6 m, W

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

Adding 2 -D vectors that form a right triangle analytically • Sketch (no scale needed) the vectors tip to tail. Draw resultant from start to finish. • Calculate the magnitude of the resultant using the Pythagorean theorem. • Calculate the angle using inverse tangent. R 24 m 18 m

Stating the resulatant angle if it is not in the first quadrant • Note: If the angle is not in the first quadrant measured from east or 0º counter-clockwise, you must adjust it. • Examples (not to scale): 50º 30º Since the angle is measured from 270º and is before 270º, subtract from 270º. Angle = 270º - 50º = 220º Since the angle is measured from 180º and is after the 180º, add to 180º. Angle = 180º + 30º = 210º

Another way to state the resultant angle if not measured ccw from East or 0º Besides 220º, this angle could be stated as 50º West of South. Note: The last direction (S) is the direction the angle is measured from. Besides 210º, this angle could be stated as 30º South of West

Vector Resolution – resolve into x and y components 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 =Ax/A so… Ax = A cos sin = Ay/A so… Ay = A sin Hint: Be sure your calculator is in degrees!

Vector Addition-Analytical To add vectors mathematically: – resolve the vectors to be added into their x- and ycomponents. – Add the x- components together to get the x component of the resultant vector (Rx ) – Add the y- components together to get the y component of the resultant vector (Ry ) – Sketch Rx and Ry tip to tail. Sketch the resultant, R from tail of Rx to tip of Ry (start to finish). – Use the pythagorean theorem and Rx and Ry to find the magnitude of resultant. – Use inverse tangent to find the angle and then adjust to find the direction of the resultant.

Vector Addition-Analytical Ex. First, calculate the x and y components of each vector. Given: Vector A is 90 u, 30 O and vector B is 50 u, 125 O. Example: Find the resultant R = A + B mathematically. By B 55 O Bx A Ay 30 O Ax Ax= 90 cos 30 O = 77. 9 u Ay= 90 sin 30 O = 45 u Bx = 50 cos 55 O = - 28. 7 u By = 50 sin 55 O = 41 u Note: Bx is negative because it is acting along the neg x axis. Using 125 O instead of the reference angle of 55 O will result in a negative value (so don’t need to add the neg).

Vector Addition-Analytical Ex. (continued) Find Rx and Ry: Rx = A x + B x R Ry Rx Ry = A y + B y Rx = 77. 9 - 28. 7 = 49. 2 u Ry = 41 + 45 = 86 u Use: R 2 = Rx 2 + Ry 2 to find magnitude of R. R 2 = (49. 2)2+(86)2 , so… R = 99. 1 u Use: = tan-1 ( Ry / Rx ) to find the angle. = tan-1 ( 86 / 49. 2 ) = 60. 2 O

Stating the final answer • All vectors must be stated with a magnitude and direction. • Angles must be adjusted to be measured from East (or 0° or the + x-axis) counterclockwise or by adding compass directions ( i. e. N of W) • When added graphically, the length of the resultant must be multiplied by the scale factor to find the magnitude.