**Problem statement:**

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

- The cross product
**A**×**B**. - The area of the parallelogram spanned by
**A**and**B**. - The
*y*and*z*components of a vector**C**= 2**i**+ C_{y }**j**+C_{z}**k**parallel 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 cross product of two vectors **is a vector** given by the following determinant:

And substituting the components of **A** and **B**:

**The cross product is always perpendicular to both vectors**. You can verify it by performing the dot product of each vector and the result of their cross product. Both have to be zero.

If we call **D** = **A**×**B**:

And doing the same for vector **B**:

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The area of the parallelogram spanned by **A** and **B** is the magnitude (modulus) of their cross vector:

If vector **C** is parallel to **B** their cross product must be zero:

And expanding the previous determinant:

But for a vector to be zero its components must be zero too:

Therefore, **C** expressed in unit vector notation is given by: