If the scalar product of two nonzero vectors is zero, what can you conclude about their relative directions?

The dot product, also known as the scalar product, is a mathematical operation that combines two vectors into a single number, a scalar. It's defined by multiplying the magnitudes of the two vectors and the cosine of the angle between them. The formula for the dot product is:
\[\begin{equation}\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos(\theta)\end{equation}\]
where \( \mathbf{a} \) and \( \mathbf{b} \) represent the vectors, \( |\mathbf{a}| \) and \( |\mathbf{b}| \) their magnitudes, and \( \theta \) the angle in between. This calculation tells us not only the size but also the direction of one vector in relation to another. The result can indicate whether two vectors are pointing in roughly the same direction (positive dot product), in opposite directions (negative dot product), or are orthogonal with zero as the outcome.

In the context of vectors, 'magnitude' refers to the vector's length or size. It's a measure of how much impact the vector has in its direction of application. For instance, the magnitude of a force vector would tell us how strong the force is. Mathematically, the magnitude of a vector \( \mathbf{v} \) is calculated by taking the square root of the sum of the squares of its components, expressed as:
\[\begin{equation}|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}\end{equation}\]
Figuratively imagining a right triangle with sides corresponding to the vector's components, the magnitude would be the length of the hypotenuse. It's crucial in many physical applications, such as when you're calculating velocity, acceleration, or any quantity that has both a direction and a magnitude.

Vectors are said to be orthogonal - a term stemming from Greek origins meaning 'right angle' - if they meet at a ninety-degree angle. This property is particularly significant when it comes to the dot product. Indeed, the dot product of two nonzero orthogonal vectors is zero because the cosine of a ninety-degree angle is zero. Two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal if, and only if,
\[\begin{equation}\mathbf{a} \cdot \mathbf{b} = 0\end{equation}\]
In a three-dimensional space, orthogonal vectors are critical for defining coordinate systems, such as Cartesian coordinates, where the unit vectors along the x, y, and z axes are all orthogonal to one another. When two vectors are orthogonal, they have no impact on each other in the sense that one does not contribute to the other’s direction or magnitude.