Dot product problems with solution

Problem statement:

Given the vectors: A = 3i + 2jk and B = 5i +5j, find:

  1. The dot product AB.
  2. The projection of A onto B.
  3. The angle between A and B.
  4. A vector of magnitude 2 in the XY plane perpendicular to B.

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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).


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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:

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