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To solve this exercise regarding the tetrahedron \(ABCD\) and the point \(M\), let’s break down the problem step by step.

### Part a) Express vector \(\overrightarrow{BM}\) as a function of vectors \(\overrightarrow{BC}\) and \(\overrightarrow{BD}\)

We are given:
\[
\overrightarrow{AM} = 3\overrightarrow{AB} - \overrightarrow{AC} - \overrightarrow{AD}.
\]
We need to express \(\overrightarrow{BM}\) in terms of \(\overrightarrow{BC}\) and \(\overrightarrow{BD}\).

First, using the relationship \(\overrightarrow{BM} = \overrightarrow{BA} + \overrightarrow{AM}\), and knowing that:
\[
\overrightarrow{BA} = -\overrightarrow{AB},
\]
we substitute \(\overrightarrow{AM}\) into this equation:
\[
\overrightarrow{BM} = -\overrightarrow{AB} + (3\overrightarrow{AB} - \overrightarrow{AC} - \overrightarrow{AD}).
\]
Simplifying this expression:
\[
\overrightarrow{BM} = (3\overrightarrow{AB} - \overrightarrow{AB}) - \overrightarrow{AC} - \overrightarrow{AD},
\]
\[
\overrightarrow{BM} = 2\overrightarrow{AB} - \overrightarrow{AC} - \overrightarrow{AD}.
\]

Next, we express \(\overrightarrow{AB}\), \(\overrightarrow{AC}\), and \(\overrightarrow{AD}\) in terms of \(\overrightarrow{BC}\) and \(\overrightarrow{BD}\):
- \(\overrightarrow{AB} = -\overrightarrow{BA} = \overrightarrow{BC} + \overrightarrow{BD}\),
- \(\overrightarrow{AC} = \overrightarrow{BC}\),
- \(\overrightarrow{AD} = \overrightarrow{BD}\).

Substituting these into the equation for \(\overrightarrow{BM}\):
\[
\overrightarrow{BM} = 2(\overrightarrow{BC} + \overrightarrow{BD}) - \overrightarrow{BC} - \overrightarrow{BD}.
\]
Simplifying this expression:
\[
\overrightarrow{BM} = 2\overrightarrow{BC} + 2\overrightarrow{BD} - \overrightarrow{BC} - \overrightarrow{BD},
\]
\[
\overrightarrow{BM} = \overrightarrow{BC} + \overrightarrow{BD}.
\]

Thus, the vector \(\overrightarrow{BM}\) is expressed as:
\[
\overrightarrow{BM} = \overrightarrow{BC} + \overrightarrow{BD}.
\]

### Part b) Determine which plane containing a face of the tetrahedron the point \(M\) lies in

We are given that:
\[
\overrightarrow{AM} = 3\overrightarrow{AB} - \overrightarrow{AC} - \overrightarrow{AD}.
\]
By analyzing this equation, note that it expresses \(M\) as a linear combination of the vectors \(\overrightarrow{AB}\), \(\overrightarrow{AC}\), and \(\overrightarrow{AD}\).

This means that point \(M\) lies on the plane formed by points \(A\), \(C\), and \(D\), which is the face of the tetrahedron \(ACD\).

### Final Answers:
- **Part a)**: \(\overrightarrow{BM} = \overrightarrow{BC} + \overrightarrow{BD}\).
- **Part b)**: The point \(M\) lies on the plane containing the face \(ACD\) of the tetrahedron.

Let me know if you need more details or have any questions!

Here are 5 related questions:
1. How can we express \(\overrightarrow{AM}\) in terms of vectors \(\overrightarrow{BC}\) and \(\overrightarrow{BD}\)?
2. How do we find the equation of the plane passing through points \(A\), \(C\), and \(D\)?
3. Can we express \(\overrightarrow{BM}\) in terms of different vector bases of the tetrahedron?
4. How does the position of point \(M\) change if the coefficients in \(\overrightarrow{AM}\) are altered?
5. How would the solution change if the given point \(M\) was defined differently, say as \(AM = 2\overrightarrow{AB} + \overrightarrow{AC} - \overrightarrow{AD}\)?

**Tip**: When dealing with tetrahedrons, a useful strategy is to express vectors in terms of multiple bases (like \(\overrightarrow{BC}\) and \(\overrightarrow{BD}\)) to simplify the problem geometry.

ABCD est un tétraèdre et M est le point défini par : AM = 3AB - AC - AD. a) Exprimer le vecteur BM en fonction des vecteurs BC et BD. b) À quel plan contenant une face du tétraèdre le point M appartient-il ?

This problem involves vector geometry within a tetrahedron ABCD. You'll learn how to express the vector BM in terms of BC and BD, and how to determine the plane containing point M.

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