The image provides a geometric representation with three non-coplanar vectors: $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. We are tasked with expressing the other vectors in the diagram in terms of these three vectors.

Step-by-step solution:

1. Understanding the Diagram:

   • The three given vectors are non-coplanar, meaning they span a three-dimensional space.
   • The vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ appear to define the edges of a parallelepiped.
   • The points marked include $O$ (the origin), and points along various lines, such as $P$, $Q$, $R$, etc.

2. Key relationships in the figure:

   • $\mathbf{OA} = \mathbf{a}$
   • $\mathbf{OB} = \mathbf{b}$
   • $\mathbf{OC} = \mathbf{c}$

3. Expressing other vectors:

   • Vector $\mathbf{OP}$: Since $P$ lies on the same line as $B$, we express $\mathbf{OP}$ as some scalar multiple of $\mathbf{b}$. Therefore, $\mathbf{OP} = k\mathbf{b}$, where $k$ is some scalar.

   • Vector $\mathbf{OQ}$: $Q$ seems to lie along the plane spanned by $\mathbf{a}$ and $\mathbf{b}$. Thus, $\mathbf{OQ}$ is a linear combination of $\mathbf{a}$ and $\mathbf{b}$, i.e., $\mathbf{OQ} = m\mathbf{a} + n\mathbf{b}$.

   • Vector $\mathbf{OR}$: Since $R$ appears to be a point in the interior of the parallelepiped, it can be expressed as a combination of $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. So, $\mathbf{OR} = p\mathbf{a} + q\mathbf{b} + r\mathbf{c}$.

   • Vector $\mathbf{OU}$: Similarly, $U$ seems to be a point involving a combination of all three vectors. We can express $\mathbf{OU} = s\mathbf{a} + t\mathbf{b} + u\mathbf{c}$.

Conclusion:

Each vector can be expressed as:

• $\mathbf{OP} = k\mathbf{b}$
• $\mathbf{OQ} = m\mathbf{a} + n\mathbf{b}$
• $\mathbf{OR} = p\mathbf{a} + q\mathbf{b} + r\mathbf{c}$
• $\mathbf{OU} = s\mathbf{a} + t\mathbf{b} + u\mathbf{c}$

Please confirm the relationships or let me know if additional clarification is needed on any of the vectors.

Would you like further details or clarifications on any step?

Related questions:

1. How can we use the cross product to verify non-coplanarity of vectors?
2. What conditions would make three vectors coplanar?
3. How can you find the volume of the parallelepiped formed by vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$?
4. How do scalar multiples of vectors change their magnitude and direction?
5. How would you solve for the scalars $p, q, r$ given specific points on the parallelepiped?

Tip:

Understanding how to express vectors in terms of a basis helps in analyzing vector spaces and geometric objects in higher dimensions!