Position & Displacement Vectors (DP IB Applications & Interpretation (AI)): Revision Note

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

  • 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

Position Vectors

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

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

  • 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