IB DP Maths: AA HL復習筆記3.9.2 Position & Displacement Vectors
Adding & Subtracting Vectors
How are vectors added and subtracted numerically?
- To?add?or?subtract?vectors numerically simply add or subtract each of the corresponding components
- In?column vector?notation just add the top, middle and bottom parts together
In?base vector?notation add each of the?i,?j, and?k?components together separately
- For example: (2i?+?j?– 5k)?– (i?+ 4j?+ 3k) = (i?– 3j?– 8k)
How are vectors added and subtracted geometrically?
- Vectors can be?added?geometrically by joining the end of one vector to the start of the next one
- The?resultant?vector will be the shortest route from the start of the first vector to the end of the second
- A?resultant?vector is a vector that results from?adding?or?subtracting?two or more vectors
- If the two vectors have the same?starting position, the second vector can be?translated?to the end of the first vector to find the resultant vector
- This results in a?parallelogram?with the resultant vector as the diagonal
- ?To?subtract?vectors, consider this as?adding on the negative vector
- For example:?a?–?b?=?a?+ (-b)
- The end of the?resultant vector a – b?will not be anywhere near the end of the vector?b
- Instead, it will be at the point where the end of the vector?-b?would be
Exam Tip
- Working in column vectors tends to be easiest when adding and subtracting
- in your exam, it can help to convert any vectors into column vectors before carrying out calculations with them
- If there is no diagram, drawing one can be helpful to help you visualise the problem
Worked Example
Position Vectors
What is a position vector?
- A position vector describes the?position?of a point in relation to the?origin
The origin is always denoted O
- The individual components of a position vector are the coordinates of its end point
- For example the point with coordinates (3, -2, -1) has position vector 3i?– 2j?–?k
Worked Example
Determine the position vector of the point with coordinates (4, -1, 8).
Displacement Vectors
What is a displacement vector?
- A?displacement vector?describes the shortest route between any two points
- It describes the?direction?and the?distance?between any two points
Exam Tip
- In an exam, sketching a quick diagram can help to make working out a displacement vector easier
Worked Example
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