Chapter 3: 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 Vector  A quantity with magnitude & direction. Scalar  A quantity with magnitude only. Here, we mainly deal with Displacement  D & Velocity  v Our discussion is valid for any vector! This chapter is mostly math! 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 Axes A point in the plane is denoted as (x,y) Usually, we define a reference frame using a standard coordinate axes. (But the choice of reference frame is arbitrary & up to us!). Rectangular or Cartesian Coordinates: 2 Dimensional “Standard” coordinate axes. A point in the plane is denoted as (x,y) Note, if its convenient, we could reverse + & - ! Standard sets of xy (Cartesian or rectangular) coordinate axes - ,+ +,+ - , - + , -

Plane Polar Coordinates Trigonometry is needed to understand these! A point in the plane is denoted as (r,θ) (r = distance from origin, θ = angle from the x-axis to a line from the origin to the point). (a) (b)

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

Vector Addition, Graphical Method Addition of scalars: “Normal” arithmetic! Addition of vectors: Not so simple! Vectors in the same direction: 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 Vectors in the Same Direction

2 methods of vector addition: 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? 2 methods of vector addition: Graphical (2 methods of this also!) Analytical (TRIGONOMETRY)

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. Note! DR < D1 + D2 (scalar addition) DR = 11.2 km 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.

Graphical Method V = V1 + V2 A 2nd graphical method of adding vectors: (100% equivalent to the tail-to-tip method!) V = V1 + V2 1. Draw V1 & V2 to scale from a 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)

Mathematically, we can move vectors around 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)

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 3.2 Use a ruler & protractor to find the length of R A two part car trip: First displacement: A = 20 km due North. Second 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