give the components of the velocity vector of a boat that is moving at 40 km/hr in a direction 20◦ south of west. (assume north is in the positive y-direction.)
The vector can be described using the components. We define a coordinate system and then break the vector as the sum of vectors along the defined axis. This makes mathematical operation easier.
Consider a vector in the 2-D plane. It is inclined to the x-axis at an angle of 30 degrees. So, it is 60 degrees inclines to the y-axis. Now, we can break our vectors as the sum of vectors along these axes.
The projection of the vector on the x-axis is vector along this axis. Similarly, the projection of the vector on the y-axis it vector along that axis. Thus, we can express vectors as a sum of vectors, commonly known as components of vectors.
Note: The components of vector essentially depend on the coordinate system we choose.