Problem 9 - Algebra and Trigonometry

Question

If the vector \(\mathbf{v}\) has initial point \(P,\) what is its terminal point? $$ \mathbf{v}=\langle- 2,0,2\rangle, P(3,0,-3) $$

Step-by-Step Solution

Answer

(1, 0, -1)

Step 1: Understand the Vector Components

The vector \( \mathbf{v} = \langle -2, 0, 2 \rangle \) represents the change from the initial point \(P(3, 0, -3)\) to the terminal point. The components of \( \mathbf{v} \) are the differences in the coordinates between the terminal point and the initial point.

Step 2: Calculate Terminal Point's Coordinates

The terminal point coordinates \((x, y, z)\) can be found by adding the vector components to the initial point's coordinates \((3,0,-3)\). This gives us: \( x = 3 + (-2) = 1 \), \( y = 0 + 0 = 0 \), and \( z = -3 + 2 = -1 \).

Step 3: Write the Terminal Point

The calculated coordinates of the terminal point are: \((1, 0, -1)\).

Key Concepts

Three-Dimensional Coordinates

Three-dimensional coordinates are a vital part of vector mathematics. They consist of three values, typically represented as \( (x, y, z) \), that describe the position of a point in three-dimensional space.

Unlike two-dimensional coordinates, which only consider horizontal and vertical positions, three-dimensional coordinates add depth. This means that any point can be plotted on a 3D grid by considering three perpendicular axes: the x-axis (which runs horizontally), the y-axis (which runs vertically), and the z-axis (which typically represents depth or height).

When working with vectors in three-dimensional space, understanding the coordinate system is crucial for determining both the position of points and the direction and magnitude of vectors. Calculating the terminal point of a vector involves adjusting the initial point's coordinates by the vector's components, as demonstrated in our exercise example.

Vector Components

Vector components break down a vector into its fundamental parts—essentially what makes the vector move from one point to another.

In our exercise, the vector \( \mathbf{v} = \langle -2, 0, 2 \rangle \) is expressed through its components, which indicate movement along the x, y, and z axes respectively. Each component represents how far and in what direction the vector shifts along each axis.

More specifically:

• The x-component \( -2 \) suggests the vector shifts 2 units left along the x-axis.
• The y-component \( 0 \) indicates there is no movement along the y-axis.
• The z-component \( 2 \) shows a shift 2 units upwards along the z-axis.

Understanding these components is essential when you add them to the initial point to find the terminal point, as each component represents the directional adjustments made to reach the final position from the start.

Initial and Terminal Points

When discussing vectors, initial and terminal points are pivotal. The initial point is where the vector begins, and the terminal point is where it ends. Both points are critical in determining the vector's direction and length.

For our vector \( \mathbf{v} = \langle -2, 0, 2 \rangle \), the initial point is given as \( P(3, 0, -3) \). To find the terminal point, we need to apply the vector's components to the coordinates of the initial point.

Here's how it's done step by step:

• Add the x-component \( -2 \) to the initial x-coordinate (3), resulting in 1.
• Add the y-component \( 0 \) to the initial y-coordinate (0), which remains 0.
• Add the z-component \( 2 \) to the initial z-coordinate (-3), yielding -1.

These calculations result in the terminal point \( (1, 0, -1) \). Understanding this concept is crucial because it illustrates how vectors translate points in space, thus connecting two distinct locations.