1 Vectors
- "You can't add apples and oranges". In a strange way, this is the reason for vectors.
- We have two separate numbers \(v_1\) and \(v_2\). That pair produces a two dimensional vector \(v\)
\(v = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\)
1.1 Vector Addition
\(v = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\)
\(w = \begin{bmatrix} w_1 \\ w_2 \end{bmatrix}\)
\(v + w = \begin{bmatrix} v_1 + w_1 \\ v_2 + w_2 \end{bmatrix}\)
Subtraction of vectors follow the same idea.
1.2 Scalar Multiplication
\(2v = \begin{bmatrix} 2v_1 \\ 2v_2 \end{bmatrix}\)
1.3 Zero Vector
The sum of the vector \(-v\) and \(v\) is the zero vector. This is \(\mathbf{0}\) wich is not the same as number zero.
The vector \(\mathbf{0}\) has two components \(0\) and \(0\).
2 Linear combination
Combining addition with scalar multiplication, we form "linear combinations" of \(v\) and \(w\).
The sum of \(cv\) and \(dw\) is a linear combination of \(v\) and \(w\).
2.1 Geometrical Interpretation (2D vectors)
- Vector addition produces the diagonal of a prallelogram.
- A vector with two components corresponds to a point in the \(xy\) plane.
- The components of \(v\) are the coordinates orf the point \(x = v_1\) and \(y = v_2\). The arrow ends at this point \((v_1, v_2)\) when it starts from \((0,0)\).
2.2 Geometrical Interpretation (3D vectors)
- The vector \(v\) corresponds to an arrow in 3-space. Usually the arrow starts at the origin where the \(xyz\) axes meet and the coordinates are \((0,0,0)\).
2.3 All combinations
Assuming every \(c\), \(d\) and \(e\) are allowed:
- The combinations \(cu\) fill a line.
- The combinations \(cu + dv\) fill a plane.
- The combinations \(cu + dv + ew\) fill three-dimensional space
3 Miscellaneous
- Vectors \(\begin{bmatrix} x \\ y \end{bmatrix}\) and \(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\) can also be written as \((x,y)\) and
\((x,y,z)\). Note that they are not a row vector. Row vector are represented as \([x, y]\) instead.