Two Chapter 3

• It is sometimes valuable to find two or more vectors, whose sum is a given vector. We may be faced with the question, "Find two vectors whose sum is the vector A?"

• Any two vectors placed head-to-tail, which forms a triangle with vector A will answer this question.

• This results in an infinite number of possible combinations. Some of these are shown below.

• We can place restriction on the types of vectors to be used in our solution.
  • We can insist the two vectors must be perpendicular to each other.

• For this restriction, there is still an infinite number of possible combinations. A few of these are illustrated below.

• If we add a further restriction that one of the two vectors must be in a particular direction (say along the X-axis of a coordinate system), then only one possible solution is present.

• The possibility shown below is really the same as the one illustrated above, only we have placed the same vectors (B and C) in reverse order when placing them tip-to-tail.

• Let us assume that the magnitude of vetor B above is B, and that the magnitude of vector C above is C.
• If i is a vector of unit length (magnitude = 1) along the X-axis, then vector B can be written as

B = Bi

• Similarily, if j is a vector of unit length (magnitude = 1) along the Y-axis, Then vector C can be written as

C = Cj

• Since vectors B and C are placed tip-to-tail in the diagram above, we then can write

A = B + C

or

A = Bi + Cj

• Component Vectors - When two vectors B (Bi) and C (Cj) add to produce a third vector A, we say that B and C are components of A.
• In adding vectors in physics, we can use trigonometry if we insist on the following restrictions
  • We will only break vectors up into two component vectors.
  • The two vectors must be perpendicular to each other.
  • One of the two must be pointing in a particular direction (usually the X-axis).
• Orthogonal or Rectangular Components - component vectors that are perpendicular to each other.
  • In the diragram above, vectors B and C are orthogonal
• Mutually independent vectors - Vectors that are perpendicular to each other.
  • In this case, neither vector has a component (that meets the above three conditions) in the direction of the other.
  • They act independent of each other.
  • An example of this would be the acceleration due to gravity which acts vertically, and a velocity vector parallel to the Earth's surface. The acceleration vector cannot be broken up into a vector parallel to the Earth's surface and another vector. Therefore, the acceleration cannot cause any change in velocity parallel to the Earth's surface. Velocities parallel to the Earth's surface and acceleration due to gravity are independent of each other.
• The magnitude and direction of a vector can be determined by a knowledge of its components. Further to this, vectors can be added and subtracted by using trigonometry and the components of the vectors. I will illustrate this by solving the same problem we solved by using a scale diagram.
• A woman walks 300 m [E], then 200 m [W 20^o S] and finally 150 m [N 40^o E]. What is her displacement after she finishes?
  • All such problems can be solved by following these steps.
    • Break each vector in the problem up into its X and Y components.
    • Add all the X components together to get the total X component of the answer.
    • Add all the Y components together to get the total Y component of the answer.
    • Use Pythagorean Theorem to find the Magnitude of the answer.
    • Use tangent q to find the direction of the answer.
    • State your answer.
  • Therefore, breaking each vector into components is shown below.

• Notice that for the 300 m[E], there is no component in the north direction because north is perpendicular to east, and that makes them mutually independent.
• Notice that d_2y and d_2x are both negative. This is because these components of d₂ point in the negative Y and X directions respectively. You must put in the negative signs, or you will get an incorrect answer!
• Notice that d_3x and d_3y are both positive. This is because these components of d₃ point in the positive X and Y directions respectively. You must put in the positive signs, or you will get an incorrect answer!
• The next step is to add all of the X components together to get the total X component of the answer.

• Now we fine the sum of all of the Y components to get the total Y component of the answer.

• Pythagorean Theorem is used to find the magnitude of her displacement.

• Notice that d_xt and d_yt are drawn pointing in the positive X and Y directions in the above diagram. This is because when we added the X components to get d_xt the result was a positive number. Similarly for d_yt.
• I either of d_xt or d_yt would have been a negative number, we would have had to draw them in the opposite direction.
• Tangent q is used to find the direction of the displacement.

• Finally we state the answer.

• In the above example, the direction was reported as [E 13^o N]. This direction can also be reported as [N 77^o E], or as [bearing 13^o ]. Bearings may also be written as @ 13^o. Remember that bearings are angles measured from north clockwise around. Therefore, a direction of [E 15^o S] is reported as a bearing of 105^o.
• To review this trigonmetric method of vector addition, run Vector Addition power point or download it.

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August 6, 2020