**Problem statement:**

Given the vectors: **A** = 3**i** + 2**j** – **k** and **B** = 5**i** +5**j**, 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**.

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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: