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? •

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 no 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°, 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 • Example: 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.

Vector Addition – 1 D • 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

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 the vectors tip to tail. (no scale needed) • Calculate the magnitude of the resultant using the Pythagorean theorem. • Calculate the angle using inverse tangent. R 24 m 18 m

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