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The vector magnitude is a measure of a vector's length. For any two-dimensional vector \( \mathbf{a} = \langle x, y \rangle \), this magnitude is calculated using the Pythagorean theorem. The formula is \( \|\mathbf{a}\| = \sqrt{x^2 + y^2} \), which essentially gives us the vector's length from the origin to the point \( (x, y) \) on a coordinate plane.
For example, if our vector \( \mathbf{a} = \langle 2, -5 \rangle \), we substitute these values into the formula, as follows:

• Calculate \( x^2 = 2^2 = 4 \)
• Calculate \( y^2 = (-5)^2 = 25 \)
• Add them to get \( x^2 + y^2 = 4 + 25 = 29 \)
• Find the square root \( \|\mathbf{a}\| = \sqrt{29} \)

This gives the magnitude or the length of vector \( \mathbf{a} \), which is useful for understanding both its size and scale.

The direction of a vector tells us where it points relative to the origin. To express a vector purely by its direction, we create a unit vector—a vector with a magnitude of 1. This is achieved by dividing each component of the original vector by its magnitude.
Consider vector \( \mathbf{a} = \langle 2, -5 \rangle \). We've determined its magnitude to be \( \sqrt{29} \). The resulting unit vector is:

• Divide the x-component: \( \frac{2}{\sqrt{29}} \)
• Divide the y-component: \( \frac{-5}{\sqrt{29}} \)

This forms the unit vector \( \mathbf{u} = \left\langle \frac{2}{\sqrt{29}}, \frac{-5}{\sqrt{29}} \right\rangle \). The purpose of this transformation is to maintain the direction of the original vector while standardizing its length to one unit. In applications, unit vectors are foundational because they allow us to handle direction without the complexity of varied magnitudes.

Sometimes, we are interested in a vector that points in the exact opposite direction. To achieve this, we create the opposite of a unit vector by simply multiplying each component of the unit vector by \(-1\). This operation flips the vector's direction, keeping the same magnitude.
Starting from our previously calculated unit vector \( \mathbf{u} = \left\langle \frac{2}{\sqrt{29}}, \frac{-5}{\sqrt{29}} \right\rangle \):

• Negate the x-component: \( -\frac{2}{\sqrt{29}} \)
• Negate the y-component: \( \frac{5}{\sqrt{29}} \)

This results in the opposite unit vector \( -\mathbf{u} = \left\langle -\frac{2}{\sqrt{29}}, \frac{5}{\sqrt{29}} \right\rangle \). This vector is exactly in the opposite direction of the original unit vector, but with the same unit magnitude. It is a crucial concept, especially in navigation and physics, for determining movements or forces in the contrary direction.