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
- For example:
- 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
Find the resultant of the vectors a = 5i – 2j and b = .
What is a position vector?
- A position vector describes the position of a point in relation to the origin
- It describes the direction and the distance from the point O: 0i + 0j + 0k or
- It is different to a displacement vector which describes the direction and distance between any two points
- The position vector of point A is written with the notation a =
- 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
Determine the position vector of the point with coordinates (4, -1, 8).
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
- It is different to a position vector which describes the direction and distance from the point O: 0i + 0j or
- The displacement vector of point B from the point A is written with the notation
- A displacement vector between two points can be written in terms of the displacement vectors of a third point
- A displacement vector can be written in terms of its position vectors
- For example the displacement vector can be written in terms of and
- For position vector a = and b = the displacement vector can be written b – a
The point A has coordinates (3, 0, -1) and the point B has coordinates (-2, -5, 7). Find the displacement vector .