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Chapter 3: Vectors
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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
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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!
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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 - ,+ +,+ - , - + , -
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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)
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Equality of two vectors
2 vectors, A & B. A = B means that A & B have the same magnitude & direction.
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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
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Adding Vectors in the Same Direction
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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)
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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
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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)
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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.)
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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.
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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)
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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)
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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)
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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.
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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
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