Scalars & Vectors Learning Outcomes

Transcription

1 1 Scalars & Vectors Learning Outcomes Differentiate between scalars and vectors. Recognise quantities as either scalars or vectors. Add vectors. Use coordinates to represent and work with vectors. Decompose vectors into components. Calculate the direction of vectors. Solve problems about vectors.

2 2 Differentiate between scalars and vectors Scalars are quantities with magnitude only, e.g. distance, time, speed, temperature, mass, For scalars, only magnitude matters.

3 3 Differentiate between scalars and vectors Vectors are quantities with magnitude AND direction, e.g. displacement, velocity, acceleration, force For vectors, both magnitude and direction both matter.

4 4 Differentiate between scalars and vectors The distance between Dublin and Cork depends on the route you take. The displacement from Dublin to Cork is constant, has a particular direction, and is different to the displacement from Cork to Dublin.

5 5 Recognise Quantities as Scalars or Vectors State whether the following are scalars or vectors: 1. energy 2. width 3. area 4. weight 5. thrust 6. frequency 7. volume

6 6 Add Vectors Triangle Rule Vectors are represented by arrows. To add vectors, place the tail of one vector at the head of the other vector. This gives the resultant. e.g. Ԧa + b

7 7 Add Vectors Parallelogram Rule Alternatively, place both vectors with their tails in the same place. Make each vector one side of a parallelogram. The diagonal is the resultant vector.

8 8 Add Vectors To subtract vectors, turn the subtracted vector around before moving it. e.g. 3 Ԧc Ԧd

9 9 Add Vectors Copy the following into your copybook and find Ԧa + b for each example:

10 10 Add Vectors Copy these diagrams into your copybook and find the following resultants: 1. 2 Ԧa b 2. Ԧa + 4b Ԧa b

11 11 Add Vectors Copy the vectors below into your copy, then add: 1. Ԧa + b + Ԧc 2. b + Ԧc + Ԧa

12 12 Use Coordinates to Represent Vectors Vectors are most easily worked with using coordinate geometry. Instead of the x-y plane, we use the Ԧi-Ԧj plane. Ԧi is one unit long in the i direction. Ԧj is one unit long in the j direction.

13 13 Use Coordinates to Work With Vectors On a single coordinate plane, draw the following vectors: 1. 3Ԧi + 4Ԧj 2. 3Ԧi + Ԧj 3. 2Ԧi 2Ԧj Calculate the length of each vector. The i and j parts of a vector are called its components. Ԧi and Ԧj are called unit vectors because they are one unit long.

14 14 Use Coordinates to Work With Vectors Let Ԧx = 3Ԧi 2Ԧj and Ԧy = 4Ԧi + Ԧj 1. Plot Ԧx and Ԧy on a coordinate plane. 2. Write Ԧx + Ԧy in terms of Ԧi and Ԧj. 3. Find Ԧx + Ԧy, the magnitude (i.e. length) of Ԧx + Ԧy. 4. Investigate if Ԧx Ԧy.

15 15 Decompose Vectors into Components Given a vector with magnitude 6, making an angle of 30 o anticlockwise to the i-axis, find its components. Find the unit vector in the same direction.

16 16 Decompose Vectors into Components Given a vector with magnitude 10, making an angle of 45 o clockwise to the i-axis, find its components. Find the unit vector in the same direction.

17 17 Calculate the Direction of Vectors Find the angle that each of the following vectors makes with the i-axis: 1. Ԧi + Ԧj 2. 4Ԧi 2Ԧj 3. 2Ԧi Ԧj + 4Ԧi + 3Ԧj 4. 2 Ԧx + Ԧy, where Ԧx = 4Ԧi + 3Ԧj and Ԧy = 6Ԧi 8Ԧj

18 18 Solve Problems about Vectors 1. Given Ԧa = 4Ԧi 10Ԧj and b = 7Ԧi + 5Ԧj, find t such that Ԧa + tb is a vector pointing along the i-axis (i.e. has no j- component). 2. If m = 2Ԧi Ԧj and n = 4Ԧi + 3Ԧj, find k and l such that km + ln = 2Ԧi 6Ԧj. 3. If 11Ԧi kԧj = 5 3Ԧi + Ԧj, find two possible values for k. 4. Prove that Ԧi + 3Ԧj 6Ԧi 2Ԧj. 5. If 9Ԧi tԧj 2Ԧi + 6Ԧj, find the value of t.

19 19 Solve Problems about Vectors Ԧp is a vector of magnitude 35 cm; Ԧq is a vector of magnitude 13 cm. If tan α = 4 3 and tan β = 5 12, i. write Ԧp and Ԧq in terms of Ԧi and Ԧj. ii. Show that Ԧp + Ԧq makes an angle of 45 o to the i-axis.