VECTORS ARE QUANTITIES THAT HAVE MAGNITUDE AND DIRECTION

VECTORS ARE QUANTITIES THAT HAVE MAGNITUDE AND DIRECTION

Vector vs. Scalar • Scalar- Quantities that have magnitude (size) only, like mass, time, distance, energy, volume, and speed. • Vector- Quantities that have both magnitude and direction like displacement, velocity, acceleration, force, momentum, and fields. 2

DRAWING A VECTOR? A vector has both size and direction. Tail Tip A vector is represented on paper by an arrow drawn to scale and pointing in the direction of the action 3

Magnitude of Vectors • The best way to describe the magnitude of a vector is to measure the length of the vector. • The length of the vector is proportional to the magnitude of the quantity it represents. 4

Magnitude of Vectors A If vector A represents a displacement of three miles to the north… B Then vector B, which is twice as long, would represent a displacement of six miles to the north! 5

Equal Vectors Equal vectors have the same length and direction, and must represent the same quantity (such as force or velocity). 6

Inverse Vectors Inverse vectors have the same length, but opposite direction. A -A 7

VECTOR PROPERTIES • Commutative Property: A+B = B+A • Associative Property: (A+B)+C = A+(B+C) • Zero Property: A+(-B) = 0, if and only if, A is equal in magnitude to B and pointing in the opposite direction. • Subtraction: A - B = A + (-B) • Multiplication: 3 x A = 3 A 8

Vector angle ranges N NW quad 270 o < < 360 o W NE quad 0 < < 90 o SW quad 180 o < < 270 o S E SE quad 90 o < < 180 o 9

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ADDITION OF VECTORS • 3 Methods – Parallelogram Method- For a quick assessment. Good for concurrent forces. – Tip-to-Tail Method- Drawing vectors to scale on paper to find an answer. Use of a pencil, ruler and protractor needed. Good for displacements. – Mathematical Method- Determining an answer using trigonometry. The vectors need to be at right angles to one another. 11

PARALLELOGRAM METHOD • Arrange the vectors tail to tail in the correct direction and draw to scale. • Draw two identical vectors as the originals to form a parallelogram. • Draw in the diagonal of the parallelogram. This is your answer called a resultant. • Measure the resultant and find the angle. 12

A B R B A THE PARALLELOGRAM METHOD 13

Concurrent Forces 14

The Resultant and the Equilibrant The sum of two or more vectors is called the resultant vector. The resultant vector can replace the vectors from which it is derived. The resultant is completely canceled out by adding it to its inverse, which is called the equilibrant. 15

TIP-TO-TAIL METHOD • Arrange the scaled vectors from the tip of one to the tail of the next. • Draw the resultant from the tail of the first vector to the tip of the last vector. • Determine the magnitude of the resultant, and find the angle from the base of the resultant. Use a ruler and protractor. 16

Displacements as Vectors Direction Magnitude 17

A Scale and Ruler 18

The Protractor The obtuse angle The acute angle 19

B A R Find the Resultant Displacement A: 12 C meters 20 o East of North B: 15 meters East C: 5 meters 30 o North of West TIP-TO-TAIL METHOD 20

MATHEMATICAL METHOD for vectors at right angles • Sketch a diagram of the vectors. • Use the pythagorean theorem to determine the magnitude of the resultant. • Use the sine, cosine, or tangent function to determine the angle from the base of the resultant. 21

T. O. A. S. O. H. C. A. H. SOHCAHTOA TRIG REVIEW Pythagorean Theorem opp hyp adj 22

MATHEMATICAL METHOD A: 85 N, West B: 150 N, North B R R B Find the Resultant A 23

VECTOR COMPONENTS • Every vector has 2 components. • One component is horizontal, or x-direction. • One component is vertical, or y-direction. • The 2 components are always to each other. • Use trig functions to find them. 24

Y-Component Any Old Vector X-Component 25

HORIZONTAL COMPONENT VERTICAL COMPONENT 26

Ff FN mgcos mgsin Inclined Planes mg 27

IN SUMMARY • Vectors have magnitude and direction • 3 methods to add vectors – Parallelogram – Tip-to-Tail – Mathematical • Components are perpendicular vectors to any old vector. 28