IB DP Maths: AA HL复习笔记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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