The scalar product, also known as the dot product, is a fundamental operation in vector algebra. For two vectors $\mathbf{a} = \begin{pmatrix} a_1\\a_2 \\a_3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} b_1\\b_2 \\b_3 \end{pmatrix}$ in three-dimensional space, their scalar product is defined as:
$$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$$
This operation takes two vectors as input and produces a scalar (hence the name) as output.
The scalar product is denoted by a dot (·) between the vectors, which is why it's also called the dot product.
The scalar product possesses several important properties that make it a versatile tool in vector calculations:
$$\mathbf{v}\cdot\mathbf{v} = v_1^2 + v_2^2 + v_3^2 = \left(\sqrt{v_1^2 + v_2^2 + v_3^2}\right)^2 = ||\mathbf{v}||^2$$
Let $\mathbf{a} = \begin{pmatrix}1 \\2 \\3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 4\\ 5\\6 \end{pmatrix}$. Then:
$$\mathbf{a} \cdot \mathbf{b} = 1(4) + 2(5) + 3(6) = 4 + 10 + 18 = 32$$
Demonstrating commutativity: $\mathbf{b} \cdot \mathbf{a} = 4(1) + 5(2) + 6(3) = 4 + 10 + 18 = 32$
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