Given the vectors: A = 3i + 2j – k and B = 5i +5j, find:
- The dot product A⋅B.
- The projection of A onto B.
- The angle between A and B.
- A vector of magnitude 2 in the XY plane perpendicular 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.
The dot product of two vectors A and B expressed in unit vector notation is given by:
Remember that the dot product returns a scalar (a number).
To find the projection of A onto B we divide the dot product we have determined before by the magnitude of B (visit the page dot product for more information):
The angle between both vectors is given by the expression we derived when we defined the dot product:
In order to find a vector C perpendicular B we equal their dot product to zero.
Vector C written in unit vector notation is given by:
And the dot product is:
The previous equation is the first condition that the components of C must obey. Moreover, its magnitude has to be 2:
And substituting the condition given by the dot product:
Finally, C expressed in unit vector notation is given by:
