Vectors

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In the field of mathematics, particularly in vector algebra, a vector is a fundamental concept that represents both magnitude and direction. Vectors can be added or subtracted to form new vectors, and they can also be scaled (multiplied) by a scalar, which changes their magnitude without altering their direction. The magnitude of a vector is a measure of its length.

The key formulae for vectors are:

Vector Addition: \(\mathbf{a} + \mathbf{b} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}\)

Vector Subtraction: \(\mathbf{a} - \mathbf{b} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} - \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 - b_1 \\ a_2 - b_2 \end{pmatrix}\)

Scalar Multiplication: \(k\mathbf{a} = k \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} = \begin{pmatrix} ka_1 \\ ka_2 \end{pmatrix}\)

Magnitude of a Vector: \(\|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2}\)

Example:
Consider two vectors \(\mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\).
The sum of \(\mathbf{a}\) and \(\mathbf{b}\) is: $$\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 + (-1) \\ 3 + 4 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \end{pmatrix}$$
If we multiply vector \(\mathbf{a}\) by a scalar, say 3, we get: $$3\mathbf{a} = 3 \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 3 \times 2 \\ 3 \times 3 \end{pmatrix} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}$$
The magnitude of vector \(\mathbf{a}\) is: $$\|\mathbf{a}\| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}$$