Ch. 3: Kinematics in 2 or 3 Dimensions; Vectors

Two Dimensional Vectors Algebraic Vector Operations Outline Two Dimensional Vectors Magnitude & Direction Algebraic Vector Operations Equality of vectors Vector addition Multiplication of vectors with scalars Scalar product of two vectors (a later chapter!) Vector product of two vectors

Vectors General discussion. Here: We’ll mainly deal with Vector  A quantity with magnitude & direction. Scalar  A quantity with magnitude only. Here: We’ll mainly deal with Displacement  D & Velocity  v But, our discussion will be valid for any kind of vector! This chapter has a lot of math! Understanding it requires a detailed knowledge of trigonometry. Problem Solving A diagram or sketch is helpful & vital! I don’t see how it is possible to solve a vector problem without a diagram!

Coordinate Systems - ,+ +,+ - , - Rectangular or Cartesian Coordinates Review of “standard” coordinate axes. A point in the x-y plane is labeled (x,y) Note, if it is convenient, we could reverse + & - - ,+ +,+ - , - + , - Standard Rectangular (x-y) Coordinate Axes

Vector & Scalar Quantities  A quantity with magnitude & direction. Scalar  A quantity with magnitude only.

Equality of Two Vectors For 2 vectors, A & B, A = B means that A & B have the same magnitude & direction.

Sect. 3-2: Vector Addition, Graphical Method Addition of scalars is “Normal” arithmetic! Addition of vectors is not so simple! For 2 vectors in the same direction: We can also use simple arithmetic Example: Travel 8 km East on day 1, 6 km East on day 2. Displacement = 8 km + 6 km = 14 km East Example: Travel 8 km East on day 1, 6 km West on day 2. Displacement = 8 km - 6 km = 2 km East “Resultant” = Displacement

Adding 2 Vectors in the Same Direction:

Graphical Method For 2 vectors NOT along the same line, adding is more complicated: Example: D1 = 10 km East, D2 = 5 km North. What is the resultant (final) displacement? There are 2 Methods of Vector Addition: Graphical (there are 2 methods of this also!) Analytical (TRIGONOMETRY)

We want to find the Resultant = DR = D1 + D2 = ? For 2 vectors NOT along same line: D1 = 10 km E, D2 = 5 km N. We want to find the Resultant = DR = D1 + D2 = ? In this special case ONLY, D1 is perpendicular to D2. So, we can use the Pythagorean Theorem. = 11.2 km Note! DR < D1 + D2 (scalar addition) The Graphical Method requires measuring the length of DR & the angle θ. Do that & find DR = 11.2 km, θ = 27º N of E

That arrow is the Resultant R This example illustrates the general rules (for the “tail-to-tip” method of graphical addition). Consider R = A + B: 1. Draw A & B to scale. 2. Place the tail of B at the tip of A 3. Draw an arrow from the tail of A to the tip of B That arrow is the Resultant R (measure the length & the angle it makes with the x-axis)

Order isn’t important! Adding vectors in the opposite order gives the same result. In the example, DR = D1 + D2 = D2 + D1 Figure 3-4. Caption: If the vectors are added in reverse order, the resultant is the same. (Compare to Fig. 3–3.)

Graphical Method Continued Adding 3 (or more) vectors V = V1 + V2 + V3 Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.

A second graphical method of adding vectors: (100% equivalent to the tail-to-tip method!) V = V1 + V2 1. Draw V1 & V2 to scale from common origin. 2. Construct a parallelogram with V1 & V2 as 2 of the 4 sides. Then, the Resultant V = The diagonal of the parallelogram from the common origin (measure the length and the angle it makes with the x-axis)

So, The Parallelogram Method may also be used for the graphical addition of vectors. Figure 3-6. Caption: Vector addition by two different methods, (a) and (b). Part (c) is incorrect. A common error! Mathematically, we can move vectors around (preserving magnitudes & directions)

Sect. 3-3: Subtraction of Vectors First, Define the Negative of a Vector: -V  the vector with the same magnitude (size) as V but with the opposite direction. V + (- V)  0 Then, to subtract 2 vectors, add one vector to the negative of the other. For 2 vectors, V1 & V2: V1 - V2  V1 + (-V2)

Multiplication by a Scalar A vector V can be multiplied by a scalar c V' = cV V'  vector with magnitude cV the same direction as V If c is negative, the result is in the opposite direction.

Example R = A + B A two part car trip: First, displacement: A = 20 km due North. Then, displacement B = 35 km 60º West of North. Find (graphically) the resultant displacement vector R (magnitude & direction). R = A + B Use a ruler & protractor to find the length of R & the angle β: Length = 48.2 km β = 38.9º