91Ó°ÊÓ

The magnitude of a vector is essentially a measure of its length. Imagine you have a vector \( \mathbf{u} = \langle a, b \rangle \), residing in a 2D space. To find out how long this vector is, you use a special formula: \[ \|\mathbf{u}\| = \sqrt{a^2 + b^2} \].
This formula is reminiscent of the Pythagorean theorem you have encountered in geometry. It calculates the "hypotenuse" of the triangle formed when the vector is decomposed into its x and y components.
For example, for our vector \( \mathbf{u} = \langle 0, -2 \rangle \), its magnitude, or length, is \( \sqrt{0^2 + (-2)^2} = \sqrt{4} = 2 \) units.

• This means the vector reaches out 2 units, regardless of the direction, in the plane.
• It’s important to remember that magnitude is always non-negative.

The coordinate plane, often called the Cartesian plane, is a two-dimensional surface where we can precisely locate points and vectors using ordered pairs \( (x, y) \). It is formed by two perpendicular axes:

• The x-axis, running horizontally.
• The y-axis, running vertically.

When we represent a vector like \( \mathbf{u} = \langle 0, -2 \rangle \), we start at the origin `(0, 0)` and move according to its components:
- Move 0 units along the x-axis (literally no movement in the x direction).
- Move -2 units along the y-axis downward.
This system allows for a clear, visual representation of directions and distances.
By using a coordinate plane, we can easily visualize not only the magnitude but also the direction of vectors, and how they interact or combine with others in the plane.

The direction of a vector is crucial in defining how it relates to the coordinate axes. It's like knowing which way an arrow points. For a vector \( \mathbf{u} = \langle a, b \rangle \), this direction can usually be expressed by an angle \( \theta \), which is found using the arctangent function \( \theta = \text{atan2}(b, a) \).
This function gives an angle that lets you determine precisely where the vector points relative to the positive x-axis.
However, if your vector lies entirely on the x or y axis, you can tell the direction without using \( \text{atan2} \). For instance, our vector \( \mathbf{u} = \langle 0, -2 \rangle \) points straight down.

• This is equivalent to an angle of 270° or -90°, indicating a direct downward direction along the negative y-axis.
• Vectors that lie along the axes have angles that are typically multiples of 90°.

Understanding this direction tells us how to align or compare different vectors in space.