FIS00FP Unit 2: Vectors and their properties – examples
Example 1:
A person walks in the following pattern: 3,1 km north, then 2,4 km west and finally 5,2 km south. a) Sketch the vector diagram that
represents this motion. b) How far and c) in what direction would a bird fly in a straight line from the same starting point to the same
final point?
Solution:
a) I placed the Cartesian origin at the starting point of the person/bird’s journey and (arbitrarily) labelled each leg of the journey as
follows:
b) The bird and the person have the same displacement vector. Let’s call it 𝑑⃗. We need to find the magnitude of 𝑑⃗. First, we will
need to resolve vectors 𝐴⃗, 𝐵
⃗⃗ and 𝐶⃗ into their respective components (the displacement vector is the resultant or vector sum of 𝐴⃗,
⃗⃗ and 𝐶⃗).
𝐵
Vector 𝐴⃗ is vertical, pointing upward:
x component of vector 𝐴⃗: 𝐴𝑥 = 𝐴𝑐𝑜𝑠90° = 3,1𝑐𝑜𝑠90° = 0 𝑘𝑚
y component of vector 𝐴⃗: 𝐴𝑦 = 𝐴𝑠𝑖𝑛90° = 3,1𝑠𝑖𝑛90° = 3,1 𝑘𝑚
⃗⃗ is horizontal and pointing to the right. Its components are as follows:
Vector 𝐵
𝐵𝑥 = 𝐵𝑐𝑜𝑠0° = 2,4𝑐𝑜𝑠0° = 2,4 𝑘𝑚
𝐵𝑦 = 𝐵𝑠𝑖𝑛0° = 2,4𝑠𝑖𝑛0° = 0 𝑘𝑚
The direction of vector 𝐶⃗ makes an angle of 270° with respect to the positive x-axis:
𝐶𝑥 = 𝐶𝑐𝑜𝑠270° = 5,2𝑐𝑜𝑠270° = 0 𝑘𝑚
𝐶𝑦 = 𝐶𝑠𝑖𝑛270° = 5,2𝑠𝑖𝑛270° = −5,2 𝑘𝑚
The x component of the displacement vector:
𝑑𝑥 = 𝐴𝑥 + 𝐵𝑥 + 𝐶𝑥 = (0) + (2,4) + (0) = 2,4 𝑘𝑚
Example 1:
A person walks in the following pattern: 3,1 km north, then 2,4 km west and finally 5,2 km south. a) Sketch the vector diagram that
represents this motion. b) How far and c) in what direction would a bird fly in a straight line from the same starting point to the same
final point?
Solution:
a) I placed the Cartesian origin at the starting point of the person/bird’s journey and (arbitrarily) labelled each leg of the journey as
follows:
b) The bird and the person have the same displacement vector. Let’s call it 𝑑⃗. We need to find the magnitude of 𝑑⃗. First, we will
need to resolve vectors 𝐴⃗, 𝐵
⃗⃗ and 𝐶⃗ into their respective components (the displacement vector is the resultant or vector sum of 𝐴⃗,
⃗⃗ and 𝐶⃗).
𝐵
Vector 𝐴⃗ is vertical, pointing upward:
x component of vector 𝐴⃗: 𝐴𝑥 = 𝐴𝑐𝑜𝑠90° = 3,1𝑐𝑜𝑠90° = 0 𝑘𝑚
y component of vector 𝐴⃗: 𝐴𝑦 = 𝐴𝑠𝑖𝑛90° = 3,1𝑠𝑖𝑛90° = 3,1 𝑘𝑚
⃗⃗ is horizontal and pointing to the right. Its components are as follows:
Vector 𝐵
𝐵𝑥 = 𝐵𝑐𝑜𝑠0° = 2,4𝑐𝑜𝑠0° = 2,4 𝑘𝑚
𝐵𝑦 = 𝐵𝑠𝑖𝑛0° = 2,4𝑠𝑖𝑛0° = 0 𝑘𝑚
The direction of vector 𝐶⃗ makes an angle of 270° with respect to the positive x-axis:
𝐶𝑥 = 𝐶𝑐𝑜𝑠270° = 5,2𝑐𝑜𝑠270° = 0 𝑘𝑚
𝐶𝑦 = 𝐶𝑠𝑖𝑛270° = 5,2𝑠𝑖𝑛270° = −5,2 𝑘𝑚
The x component of the displacement vector:
𝑑𝑥 = 𝐴𝑥 + 𝐵𝑥 + 𝐶𝑥 = (0) + (2,4) + (0) = 2,4 𝑘𝑚
