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Week 1 / Review

Week 2

Scalars & Vectors

  • A scalar is magnitude only (no direction).

    Examples of scalar :
    temp, pressure, time, volume, speed (how fast but not direction), & mass.
    We use regular variables to describe them :
    - T (temperature)
    - p (pressure)
    - V (volume)
    - m (mass)
    - t (time)
    A vector is a quantity that has magnitude (how long it is) and direction. Examples of vectors :
    - Velocity
    - Force
    - Acceleration
    - Magnetic field
    - Electric field

    A ball is thrown w/ velocity of 37 m/s (it has a magnitude and direction)

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    A box is pushed(force) 10N (Newtons) (is a unit of magnitude and direction)

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    A unit vector is a vector that has the magnitude equal to 1. The unit vectors are denoted by the "cap" symbol ^.

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    i, j, and k are the unit vectors in the directions of the x-axis, y-axis, and z-axis respectively in a 3-dimensional plane. i.e.

    • Vector Multiplication

  • Vectors can be multiplied either using either dot product or cross product.

    • Dot Product

  • Given two vectors, what is the angle between them? Since vectors have no position, we are as usual free to place vectors wherever we like. If the two vectors are placed tail-to-tail, there is now a reasonable interpretation of the question: we seek the measure of the smallest angle between the two vectors, in the plane in which they lie.

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    Since the angle θ lies in a triangle, we can compute it using a bit of trigonometry, namely, the law of cosines.

    Example (A = |A|, B = |B|, & |A - B|) :

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    So a bit of simple arithmetic with the coordinates of A and B allows us to compute the cosine of the angle between them. If necessary we can use the arccosine to get θ, but in many problems cosθ turns out to be all we really need. This is called the dot product and write it as .

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    Example 1 (find the angle between the given vectors) :

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    Recall :
    |A||B|cosθ = A ∙ B


    A ∙ B = (1*3) + (2*1) + (1*-5) = 0


    |A||B| =

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    So :

    1) cosθ = 0 / (√6 * √35)

    2) cosθ = 0

    3) θ = π/2

    • Cross Product

  • Another useful operation: Given two vectors, find a third (non-zero!) vector perpendicular to the first two.

    Given (all vectors)(find vector v):

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    1)
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    2)
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    Given A and B, there are typically two possible directions and an infinite number of magnitudes that will give a vector perpendicular to both A and B. We know how to compute the magnitude of A×B; it's a bit messy but not difficult.

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    This particular answer to the problem turns out to have some nice properties, and it is dignified with a name: the cross product

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    While there is a nice pattern to this vector, it can be a bit difficult to memorize; here is a convenient mnemonic. The determinant of a two by two matrix is

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    You will have noticed that the three expressions in parentheses on the last line are precisely the three coordinates of the cross product; replacing x, y, z by i, j, k gives us

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    Here is a curious fact about this quantity that turns out to be quite useful later on: Given two vectors, we can put them tail to tail and form a parallelogram, as in figure 14.4.1. The height of the parallelogram, h, is |A|sinθ, and the base is |B|, so the area of the parallelogram is |A||B|sinθ, exactly the magnitude of |A×B|.

    Example 2 :

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