What is the expression for vector e in terms of its magnitude and direction?

The expression for vector **e** is **e = |e| * (e/|e|)**, where |e| is the magnitude and (e/|e|) is the direction.

In more detail, a vector is a quantity that has both magnitude (size) and direction. To express a vector **e** in terms of its magnitude and direction, we use the formula **e = |e| * (e/|e|)**. Here, **|e|** represents the magnitude of the vector, which is a measure of how long the vector is. The term **(e/|e|)** represents the direction of the vector, which is a unit vector. A unit vector has a magnitude of 1 and points in the same direction as the original vector.

To find the magnitude **|e|** of a vector **e** with components (x, y), you use the Pythagorean theorem: **|e| = √(x² + y²)**. This gives you a single number that tells you how long the vector is.

The direction of the vector is given by the unit vector **(e/|e|)**. To find this, you divide each component of the vector by its magnitude. For a vector **e** with components (x, y), the unit vector is **(x/|e|, y/|e|)**. This process normalises the vector, keeping its direction but scaling it down to a length of 1.

So, by combining the magnitude and the unit vector, you can fully describe the vector **e** in terms of its magnitude and direction. This is a fundamental concept in vector mathematics and is very useful in various applications, from physics to engineering.