5.4. Representation of Vectors

A vector is depicted as an arrow drawn in a coordinate plane. The tail of the vector is at the initial point, and the head of the vector is at the terminal point.

In geometry, vectors can be visually represented as arrows on a graph. The starting point of the arrow is called the 'tail', and the end point is the 'head'. The length of the arrow represents the size of the vector (its magnitude), while the direction the arrow points shows the direction of the vector. This is a foundational way to understand vectors because it gives an immediate visual indication of how large and in what direction the vector is acting.

Imagine you are walking in a park. The path you take from a bench to a playground can be represented as a vector. The distance you walk represents the magnitude of this vector, while the direction you walk in (toward the playground) represents its direction. By picturing it as an arrow on a map, you can see exactly how far you went and in which way.

In 2D, a vector 𝐴⃗ is written as: $𝐴⃗ = 𝐴_x \, 𝑖̂ + 𝐴_y \, 𝑗̂$ where 𝐴_x and 𝐴_y are the x and y components, and 𝑖̂ and 𝑗̂ are unit vectors along the x-axis and y-axis, respectively.

In a two-dimensional space, a vector can also be expressed as the sum of its two components - one that acts along the horizontal (x-axis) and one along the vertical (y-axis). The notation $𝐴⃗ = 𝐴_x \, 𝑖̂ + 𝐴_y \, 𝑗̂$ indicates that the vector 𝐴⃗ has a component 𝐴_x in the x-direction and a component 𝐴_y in the y-direction. Here, 𝑖̂ and 𝑗̂ are unit vectors that point strictly in the x and y directions, with a length of one unit.

Think of navigating through a city on a map. If you want to go to your friend's house, you might take a route that involves moving east and then north. You could describe your journey as moving a certain number of blocks east (the x-component) and then a certain number of blocks north (the y-component). This represents how we break down the distance and direction into clear, manageable parts.

In 3D, a vector 𝐴⃗ is written as: $𝐴⃗ = 𝐴_x \, 𝑖̂ + 𝐴_y \, 𝑗̂ + 𝐴_z \, 𝑘̂$ where 𝐴_x, 𝐴_y, and 𝐴_z are the components along the x, y, and z axes, and 𝑘̂ is the unit vector along the z-axis.

In three-dimensional space, vectors can be represented similarly, but now we also account for movement up or down. The formula $𝐴⃗ = 𝐴_x \, 𝑖̂ + 𝐴_y \, 𝑗̂ + 𝐴_z \, 𝑘̂$ breaks the vector into its three components: 𝐴_x (x-direction), 𝐴_y (y-direction), and 𝐴_z (z-direction). The unit vector 𝑘̂ represents direction in the z-axis, just like 𝑖̂ and 𝑗̂ represent the x and y directions respectively.

Consider flying an airplane. To describe the airplane's flight path, you need to take into account not just how far it travels east or west (x-axis) and how far it travels north or south (y-axis), but also how high or low it goes (z-axis). Each component of the vector corresponds to one of these dimensions, giving a complete description of the airplane's trajectory.

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