Vector BasicsVector Basics

OBJECTIVESOBJECTIVES

CONTENT OBJECTIVE: TSWBAT read CONTENT OBJECTIVE: TSWBAT read and discuss in groups the meanings and and discuss in groups the meanings and differences between Vectors and Scalarsdifferences between Vectors and Scalars

LANGUAGE OBJECTIVE: TSW read and LANGUAGE OBJECTIVE: TSW read and discuss the key vocabulary words Vectors discuss the key vocabulary words Vectors and Scalars and Resultants. and Scalars and Resultants.

ScalarScalar

Scalar quantities only Scalar quantities only have magnitude (size have magnitude (size represented by a number represented by a number & unit) ex. Include mass, & unit) ex. Include mass, volume, time, speed, volume, time, speed, distancedistance

VectorsVectors

Vector quantities have Vector quantities have magnitude and direction. magnitude and direction. Ex. Include force, Ex. Include force, velocity, acceleration, velocity, acceleration, displacementdisplacement

Vector RepresentationVector Representation

Vectors are represented Vectors are represented as arrowsas arrows

The length of the vector The length of the vector represents the magnituderepresents the magnitude

tail head or tipstart end

Vectors are always drawn to a Vectors are always drawn to a scale comparing the magnitude of scale comparing the magnitude of

your vector to the metric scaleyour vector to the metric scale

Ex.Ex. 1.0 cm = 1.0 m/s1.0 cm = 1.0 m/s

Draw a 3.0 m/s East Draw a 3.0 m/s East vectorvector

Vector will be 3.0 cm Vector will be 3.0 cm longlong

Ex.Ex. 1.0 cm = 1.5 m/s1.0 cm = 1.5 m/s

Draw a 3.0 m/s East Draw a 3.0 m/s East vectorvector

Vector will be 2.0 cm Vector will be 2.0 cm longlong

Direction of VectorsDirection of VectorsDirection of vectors is represented by the Direction of vectors is represented by the way the arrow is pointedway the arrow is pointed

Vector components are based on Vector components are based on coordinate plane so vectors can point in coordinate plane so vectors can point in negative or positive directionsnegative or positive directions

Positive X, Positive YNegative X, Positive Y

Negative X, Negative Y Positive X, Negative Y

N

E

S

W

Resultant VectorsResultant Vectors

A resultant vector is A resultant vector is produced when two or more produced when two or more vectors combinevectors combine If vectors are at an angle, If vectors are at an angle, vectors are always drawn tip vectors are always drawn tip to tailto tail

Adding and subtracting vectors Adding and subtracting vectors –– Same DirectionSame Direction

If the vectors are equal in If the vectors are equal in direction, add the quantities to direction, add the quantities to each other.each other.

Example:Example:

Adding and subtracting vectors Adding and subtracting vectors –– Opposite DirectionsOpposite Directions

If the vectors are exactly If the vectors are exactly opposite in direction, subtract opposite in direction, subtract the quantities from each other.the quantities from each other.

Example:Example:

Vectors at Right Angles to each Vectors at Right Angles to each otherother

If vectors act at right angles to If vectors act at right angles to each other, the resultant each other, the resultant vector will be the hypotenuse vector will be the hypotenuse of a right triangle.of a right triangle.

Use Pythagorean theorem to Use Pythagorean theorem to find the resultantfind the resultant

Hypotenuse = resultant vector

Pythagorean TheoremPythagorean Theoremaa2 2 + b+ b2 2 = c= c2 2 where c is the resultantwhere c is the resultant

a

b

c

Example:Example:

A hiker leaves camp and A hiker leaves camp and hikes 11 km, north and hikes 11 km, north and then hikes 11 km east. then hikes 11 km east. Determine the resulting Determine the resulting displacement of the hiker.displacement of the hiker.

112 + 112 = R2

121 + 121 = R2

242 = R2

R = 15.56 km, northeast

Calculating a resultant Calculating a resultant vectorvector

If two vectors have known If two vectors have known magnitudes and you also know the magnitudes and you also know the measurement of the angle (measurement of the angle (θθ) ) between them, we use the following between them, we use the following equation to find the resultant vector.equation to find the resultant vector.

RR22 = A = A22 + B + B22 – 2ABcosθ – 2ABcosθ

Use this for angles other than 90º

Make sure your calculator is set to DEGREES!

(go to MODE)

Example 1:Example 1:

θθ = 110° = 110°

RR22 = 5.0 = 5.022 + 4.0 + 4.022 – 2(5.0)(4.0)(cos 110) – 2(5.0)(4.0)(cos 110)

RR22 = 54.68 = 54.68

R = 7.39 N, SouthwestR = 7.39 N, Southwest

R

θθ

RR2 2 = 4.3= 4.322 + 5.1 + 5.122 – (2)(4.3)(5.1)(cos – (2)(4.3)(5.1)(cos 35)35)

RR22 = 8.57 = 8.57

R = 2.93 m, northwestR = 2.93 m, northwest

θθ = 35º

θθ

4.3 m

5.1 m

R

Example 2:Example 2:

Vector EquationsVector Equations

Pythagorean TheoremPythagorean Theoremaa2 2 + b+ b2 2 = c= c2 2 where c is the where c is the resultantresultant

Law of Cosines Law of Cosines RR22 = A = A22 + B + B22 – 2ABcosθ – 2ABcosθ