To show that any vector $\mathbf{v}$ can be written as $\left| \mathbf{v} \right| \cos \theta$ and $\left| \mathbf{v} \right| \sin \theta$, let's go through the reasoning step-by-step.

1. Define the Vector Components: Let $\mathbf{v}$ be a vector in a two-dimensional Cartesian plane, with components $(v_x, v_y)$.

2. Magnitude of the Vector: The magnitude (or length) of $\mathbf{v}$, denoted $\left| \mathbf{v} \right|$, is given by: $\left| \mathbf{v} \right| = \sqrt{v_x^2 + v_y^2}$

3. Angle with the Positive x-Axis: Suppose $\theta$ is the angle that $\mathbf{v}$ makes with the positive $x$-axis. By trigonometry, we can express the components of $\mathbf{v}$ in terms of $\theta$ and $\left| \mathbf{v} \right|$: $v_x = \left| \mathbf{v} \right| \cos \theta$ $v_y = \left| \mathbf{v} \right| \sin \theta$

4. Expressing $\mathbf{v}$ in Terms of Magnitude and Direction: Since $\mathbf{v} = (v_x, v_y)$, we can substitute the expressions for $v_x$ and $v_y$: $\mathbf{v} = \left( \left| \mathbf{v} \right| \cos \theta, \left| \mathbf{v} \right| \sin \theta \right)$

Thus, any vector $\mathbf{v}$ can indeed be written in the form $\left| \mathbf{v} \right| \cos \theta$ and $\left| \mathbf{v} \right| \sin \theta$, where $\left| \mathbf{v} \right|$ is the magnitude of the vector, and $\theta$ is the angle it makes with the positive $x$-axis.

Would you like more details on any step or further clarification? Here are some related questions:

1. How can this representation be extended to three dimensions?
2. Why is the magnitude important in vector representation?
3. How does this representation help in vector addition?
4. Can we represent vectors in terms of other angles besides the x-axis angle?
5. What happens to this form if the angle $\theta$ is 90°?

Tip: Understanding vector components in terms of trigonometric functions helps in analyzing directional quantities, like forces and velocities, in physics.