Vectors Directional Physics

Vectors Many quantities in Physics have a direction associated with them: Force can be 20.0 N [W], displacement can be 2.00 m [N], velocity can be 12 m/s [SW]. Scalars are measures without direction: Energy in Joules, Power in Watts, distance, speed. Vector quantities are expected to have directions stated with them so on tests, etc. you will need to ensure this is done. The directions are assigned based on the page you write on: North is to the top of the page, East to the right and so on. Directions like 30.0o N of E or E 30.0o N are possible as well.

Vectors Vectors can also be represented as diagrams, with specific rules. An arrow is used to draw a vector, the initial point is the tail of the vector and the terminal point is the tip of the arrow. The orientation of the arrow indicates its direction, and the length of the arrow indicates its magnitude (size or measurement). 3.0 m [N] 6.0 m [E]

Vectors Vector quantities can be denoted with an arrow over the variable: 𝐹 , 𝑑 , 𝑣 represent Force, displacement, velocity Scalar quantities can be shown without the arrow d, v, E as distance, speed or Energy. Bars are used if you wish to represent the magnitude of a vector quantity. | 𝐹 |, | 𝑑 |, | 𝑣 | as magnitude of Force, displacement, velocity Note: texts will frequently omit the arrow for vectors so whether it’s a vector or a scalar will have to be determined based on the context of the question.

Adding Vectors When adding two dimensional vectors (like North and South added together), we use a default that West and South are negative quantities. Ex: 3.0 m [E] + 5.0 m [W] = 3.0 m – 5.0 m = - 2.0 m = 2.0 m [W] There are special rules for adding vectors, that are not linear. Vector diagrams are required for these types of questions. (Note: They could be requested for linear problems as well). The rule for adding any vector is to add the tip of one vector to the tail of the other vector. (tip-to-tail). The resultant (net) of the these two vectors is found by completing the diagram (usually a triangle) and solving the triangle with Trigonometry. Note: Subtraction is adding a negative vector. A negative vector has its direction switched.

Adding Vectors Example If 3.0 m [N] is added to 4.0 m [E] then the diagram looks like: The resultant in this case is the hypoteneuse. It is drawn as a dotted line and ends up so the two tips connect. Note the 4.0 m vector is LONGER. As this is the resultant displacement, we use the symbol dR. (We would use vR and FR for those quantities). The magnitude of the resultant (or net) displacement is found using Pythagorean Theorem, the angle is also needed (where two tails are). 3.0 m 4.0 m dR

Adding Vectors Example |dR| = [(4.0 m)2 + (3.0 m)2]½ = 5.0 m (makes sense as it’s the hypoteneuse) All vectors have directions, so the angle of the dotted line is needed, and it must be at the tail of the vector to give a proper direction. Find θ. Tan θ = 4/3, θ = tan-1 (4/3) = 53o dR = 5.0 m [N 53o E] or [53o E of N] Note all numbers rounded to 2 sig digs as this was least number of sig digs in the provided numbers. 3.0 m 4.0 m dR θ

Adding Vectors Linear Example Adding 2.0 m/s [E] to 5.0 m/s [W] with diagrams is a little different but the rules are still followed: vR = 2.0 m/s + (- 5.0 m/s) = - 3.0 m/s = 3.0 m/s [W] Note: The least number of decimals rule is used here to round addition or subtraction answers. vR 2.0 m/s 5.0 m/s

Vector Problems 1) Find the resultant displacement of 10. m [N] and 15 m [E]. 2) If u = 15 m/s [S] and v = 20. m/s [E] and w = 12 m/s [W], find: a) u + v b) u – v c) 2v + u d) 3w – 2v e) 2v - u Answers: 1) 18 m [56o E of N] 2a) 25 m/s [53o E of S] b) 25 m/s [53o W of S] c) 43 m/s [21o S of E] d) 76 m/s [W] e) 43 m/s [21o N of E]

Who uses Vectors? Airplanes – directions and headings Boats – directions and headings Engineers – directions of all the forces acting on a beam for support Architects – can a design stand on its own with the forces acting on stress points? Orienteering (geocaching)

3U Textwork on Text sheet IB Textwork – Giancoli 7Page 67 Q#1, 2, 3, 5 – 9/ MisQ #1 – 3/ Page 68 P#1 – 14