Vector Basics

OBJECTIVES CONTENT OBJECTIVE: TSWBAT read and discuss in groups the meanings and differences between Vectors and Scalars LANGUAGE OBJECTIVE: TSW read and discuss the key vocabulary words Vectors and Scalars and Resultants.

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

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

Vector Representation Vectors are represented as arrows Vectors are represented as arrows The length of the vector represents the magnitude The length of the vector represents the magnitude tail head or tip startend

Vectors are always drawn to a scale comparing the magnitude of your vector to the metric scale Ex. 1.0 cm = 1.0 m/s 1.0 cm = 1.0 m/s Draw a 3.0 m/s East vector Vector will be 3.0 cm long Ex. 1.0 cm = 1.5 m/s 1.0 cm = 1.5 m/s Draw a 3.0 m/s East vector Vector will be 2.0 cm long

Direction of Vectors Direction of vectors is represented by the way the arrow is pointed Vector components are based on coordinate plane so vectors can point in negative or positive directions Positive X, Positive YNegative X, Positive Y Negative X, Negative Y Positive X, Negative Y N E S W

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

Adding and subtracting vectors – Same Direction If the vectors are equal in direction, add the quantities to each other. Example:

Adding and subtracting vectors – Opposite Directions If the vectors are exactly opposite in direction, subtract the quantities from each other. Example:

Vectors at Right Angles to each other If vectors act at right angles to each other, the resultant vector will be the hypotenuse of a right triangle. Use Pythagorean theorem to find the resultant

Hypotenuse = resultant vector Pythagorean Theorem a 2 + b 2 = c 2 where c is the resultant a b c

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

= R = R = R 2 R = km, northeast

Calculating a resultant vector If two vectors have known magnitudes and you also know the measurement of the angle (θ) between them, we use the following equation to find the resultant vector. R 2 = A 2 + B 2 – 2ABcosθ Use this for angles other than 90º Make sure your calculator is set to DEGREES! (go to MODE)

Example 1: θ = 110° R 2 = – 2(5.0)(4.0)(cos 110) R 2 = R = 7.39 N, Southwest R θ

R 2 = – (2)(4.3)(5.1)(cos 35) R 2 = 8.57 R = 2.93 m, northwest θ θ = 35º θ 4.3 m 5.1 m R Example 2:

Vector Equations Pythagorean Theorem a 2 + b 2 = c 2 where c is the resultant Law of Cosines R 2 = A 2 + B 2 – 2ABcosθ