Vector Component Form

We have seen how we can represent vectors as a combination of standard unit vectors to turn geometric operations like vector addition, subtraction, and scalar multiplication into algebraic ones. Recall that we also learned that vectors can be defined by the position of their head when the tail is placed on the origin, which we called position vectors.

Now we can combine these idea with the previous one to explore a new way of representing vectors: by the coordinates of the point which defines the position vector. When we do this, we are able to work with vectors in a purely algebraic way, which will later allow us to use more advanced algebraic techniques to solve interesting vector problems.

For example, we can write a vector with it's head at the point $(4, 3)$ in column vector form as

$$\begin{bmatrix} 4 \\ 3 \end{bmatrix}$$

Recall that this vector is also equal to $4\underset{\sim}{i} + 3\underset{\sim}{j}$, and since these standard unit vectors have their heads at $(1, 0)$ and $(0, 1)$ respectively, we can also write

$$\begin{bmatrix} 4 \\ 3 \end{bmatrix} = 4 \begin{bmatrix} 1 \\ 0 \end{bmatrix} + 3 \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$

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