The statement is false.

Here’s why:

Given the vector equation $\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c}$ with $\mathbf{a} \neq \mathbf{0}$, we cannot necessarily conclude that $\mathbf{b} = \mathbf{c}$. The dot product between two vectors is related to both the magnitude of the vectors and the angle between them. It is possible for the dot products $\mathbf{a} \cdot \mathbf{b}$ and $\mathbf{a} \cdot \mathbf{c}$ to be equal, even if $\mathbf{b} \neq \mathbf{c}$, provided the vectors $\mathbf{b}$ and $\mathbf{c}$ differ in a way that still yields the same projection along $\mathbf{a}$.

For example: If $\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c}$, $\mathbf{b}$ and $\mathbf{c}$ could be distinct vectors, but both could have the same projection on $\mathbf{a}$, thus making the dot products equal without requiring $\mathbf{b} = \mathbf{c}$.

Would you like a more detailed explanation?

---

Related Questions:

1. What are the properties of the dot product that affect vector equality?
2. Can two vectors have the same dot product with another vector but still be different?
3. How does the angle between vectors influence the dot product?
4. What is the geometric interpretation of the dot product?
5. How can the dot product be used to find the projection of one vector onto another?

Tip: The dot product of two vectors is zero when they are orthogonal (perpendicular).