A vector and its representation

Let's start by defining what a vector is. Can anyone tell me what characteristics define a vector?

Isn't it true that a vector has both magnitude and direction?

Exactly! A vector is represented by an arrow, where the length signifies its magnitude and the arrow’s direction indicates its orientation. This means a vector is often depicted visually, like in figure representations.

How do we usually write the components of a vector?

Great question! The components of a vector can be expressed in a column form, typically denoted by $\vec{v} = [v_1, v_2, v_3]$. Remember, these components depend on the coordinate system we are using. As a memory aid, you can think of 'C' for 'Coordinate' when considering each component 'v'.

What happens if we change the coordinate system?

When we change the coordinate system, the representation of the vector changes, but the vector itself—its direction and magnitude—remains the same! This independence from the coordinate system is crucial in vector analysis.

So, it’s like changing the view of a sculpture; it looks different from various angles but is the same sculpture?

Exactly, that’s a perfect analogy! Let's summarize: a vector is characterized by magnitude and direction, represented as an arrow, and its components are written in a specific coordinate system. When changing coordinates, its representation alters, but its intrinsic properties do not.

Now, let’s dive deeper into how we determine a vector's components along different basis vectors. Who can remind us how we calculate a component along a basis vector?

We use dot products to find a vector's component along a basis vector.

Exactly! For instance, the component of vector $\vec{v}$ along basis vector $e_i$ is given by: $v_i = \vec{v} \cdot e_i$.

Are there any special considerations when moving between coordinate systems?

Good point! While the physical vector remains unchanged, its representation as a matrix or column vector can differ depending on the coordinate system used. It's crucial to visualize this change and recognize that the underlying vector doesn't change, similar to how you might describe a location differently based on maps of varying scales.

So, will the numeric values of the components be different?

Yes, they often will be, especially in rotated coordinate systems. Just remember the core principle: the vector itself exists independently of how we choose to represent it!

I think I understand now! It all comes back to the fact that the representation is like the clothing we put on; it can change, but who we are remains the same!

Exactly! To recap, the orientation and magnitude of a vector remain unchanged despite its representation varying with different coordinate systems. Keep practicing these concepts, as they’ll be foundational for our next discussions!