Given the vectors: A = 3i + 2j – k and B = 5i +5j.
Determine:
- Their magnitude.
- The direction of B.
- A + B
- A -2 B
- A unit vector parallel to A.
- A vector of magnitude 2 and opposite to B
Solution:
It is essential when working with vectors to use proper notation. Always draw an arrow over the letters representing vectors. You can also use bold characters to represent a vector quantity.
Vectors A and B are written using the unit vector notation.
The magnitude of A is given by:
Similarly, the magnitude of B is:
The magnitude of a vector is always a positive number.
In the figure below vector B is shown, as well as the standard unit vectors (in red).
As we saw when we first introduced the scalar and vector quantities, we can derive the polar components of a vector from the Cartesian coordinates:
The vector sum of A and B is given by:
In order to solve section (d) we multiply B by -2 and then we add A:
A unit vector is created from another one by dividing the last by its magnitude:
We can find a vector of magnitude 2 and opposite to B by multiplying a unit vector parallel to B by -2:
