Problem 3 - Algebra and Trigonometry

Question

Find the vector v with initial point \(P\) and terminal point \(Q\) . $$ P(1,-1,0), Q(0,-2,5) $$

Step-by-Step Solution

Answer

The vector \( \mathbf{v} \) is \((-1, -1, 5)\).

Step 1: Identifying Components of the Vector

To find the vector \( \mathbf{v} \) with initial point \( P = (1, -1, 0) \) and terminal point \( Q = (0, -2, 5) \), we need to calculate the differences in each coordinate.

Step 2: Calculating X-Component

Find the difference in the x-coordinates: \(x_{Q} - x_{P} = 0 - 1 = -1\).

Step 3: Calculating Y-Component

Find the difference in the y-coordinates: \(y_{Q} - y_{P} = -2 - (-1) = -2 + 1 = -1\).

Step 4: Calculating Z-Component

Find the difference in the z-coordinates: \(z_{Q} - z_{P} = 5 - 0 = 5\).

Step 5: Constructing the Vector

Combine the components to write the vector: \( \mathbf{v} = (-1, -1, 5) \).

Key Concepts

Directional Vectors

Directional vectors are an essential concept in vector computation. They help us understand the direction and magnitude of movement from one point to another. In the Cartesian coordinate system, a directional vector starts at an initial point, like point \( P \), and ends at a terminal point, like point \( Q \). The vector essentially shows the direction needed to move from \( P \) to \( Q \).
To find this vector, you calculate the difference between the terminal and initial points' coordinates.

• X-component: \( x_{Q} - x_{P} \)
• Y-component: \( y_{Q} - y_{P} \)
• Z-component: \( z_{Q} - z_{P} \)

These components form the vector \( \mathbf{v} = (x_{Q} - x_{P}, y_{Q} - y_{P}, z_{Q} - z_{P}) \), which gives us the full picture of how to move in space. By calculating each component, we essentially "translate" from one location to the next, guided by the vector's direction.

3D Vectors

3D vectors extend the principles of directional vectors into three-dimensional space. These vectors appear as ordered triples, like \( \mathbf{v} = (x, y, z) \). They capture movement not just in a plane, but in three-dimensional space.
Working with 3D vectors involves:

• Understanding both direction and magnitude in three axes (x, y, and z).
• Calculating vector components as differences across these three dimensions.

This importance is vividly seen in our example where the vector \( \mathbf{v} = (-1, -1, 5) \) describes a specific direction and distance to travel in 3D space, reflecting a deeper understanding of spatial relationships beyond flat planes.

Coordinate Geometry

Coordinate geometry allows us to visualize and compute the relationships between points, lines, and shapes in a coordinate system. Vectors are a key component of this, providing the tools to describe how points relate to each other spatially.

• This discipline focuses on how vectors translate points within spaces defined by coordinates.
• It enables calculations of distances and directions between points.

By using coordinate geometry concepts, complex spatial problems become manageable. Calculating vectors, as demonstrated, showcases how shifts and movements between points can be decomposed into simple arithmetic steps, making complex spatial reasoning accessible and understandable.