Problem 7 - Algebra and Trigonometry

Question

If the vector \(\mathbf{v}\) has initial point \(P,\) what is its terminal point? $$ \mathbf{v}=\langle 3,4,-2\rangle, P(2,0,1) $$

Step-by-Step Solution

Answer

The terminal point is \((5, 4, -1)\).

Step 1: Understand the Vector and Initial Point

The vector \( \mathbf{v} \) is given as \( \langle 3, 4, -2 \rangle \), which means it moves 3 units in the x-direction, 4 units in the y-direction, and -2 units in the z-direction from its initial point. The initial point \( P \) is given as \( (2, 0, 1) \).

Step 2: Calculate the x-coordinate of the Terminal Point

Add the x-component of the vector to the x-coordinate of the initial point: \( x_{terminal} = 2 + 3 = 5 \).

Step 3: Calculate the y-coordinate of the Terminal Point

Add the y-component of the vector to the y-coordinate of the initial point: \( y_{terminal} = 0 + 4 = 4 \).

Step 4: Calculate the z-coordinate of the Terminal Point

Add the z-component of the vector to the z-coordinate of the initial point: \( z_{terminal} = 1 - 2 = -1 \).

Step 5: Compile Terminal Point Coordinates

Combine the calculated x, y, and z coordinates to find the terminal point: \( (5, 4, -1) \).

Key Concepts

Initial Point

The initial point is where a vector starts its journey in a three-dimensional space. It's like the home base for the vector. In our example, the initial point is given as \( P(2, 0, 1) \). This means the starting coordinates are 2 in the x-axis, 0 in the y-axis, and 1 in the z-axis.
Understanding the initial point is crucial because it helps us determine where the vector begins before it moves in the specified directions. Without knowing the initial point, you wouldn't be able to accurately calculate where the vector ends up.

Terminal Point

The terminal point is where the vector lands after moving from the initial point. It marks the destination of the vector. To find the terminal point, you need to follow the vector's "route," adding each component of the vector to the corresponding coordinate of the initial point.
Using our vector \( \mathbf{v} = \langle 3, 4, -2 \rangle \) and the initial point \( P(2, 0, 1) \), the terminal point can be calculated by:

• Adding the vector's x-component (3) to the x-coordinate of the initial point (2).
• Adding the vector's y-component (4) to the y-coordinate of the initial point (0).
• Adding the vector's z-component (-2) to the z-coordinate of the initial point (1).

This results in the terminal point \( (5, 4, -1) \). So, in essence, the terminal point tells us where the vector ends up after it's done moving from its start point.

Vector Components

Vector components represent how much a vector moves in each direction in three-dimensional space. In essence, they describe the vector's motion. For a vector \( \mathbf{v} \) like \( \langle 3, 4, -2 \rangle \), the components are:

• x-component: 3 (moves 3 units along the x-axis)
• y-component: 4 (moves 4 units along the y-axis)
• z-component: -2 (moves -2 units along the z-axis)

These components dictate the vector's "path" from the initial point to the terminal point. Without the components, it would be impossible to determine the exact position or trajectory of the vector.

Coordinate Calculation

Coordinate calculation involves deriving the exact point where the vector ends, based on its components and initial point. To perform this calculation, we add each component of the vector to the corresponding coordinate of the initial point:

• For the x-coordinate: Add the x-component to the x-value of the initial point (\( 2 + 3 = 5 \)).
• For the y-coordinate: Add the y-component to the y-value of the initial point (\( 0 + 4 = 4 \)).
• For the z-coordinate: Add the z-component to the z-value of the initial point (\( 1 - 2 = -1 \)).

After these calculations, you determine the terminal point to be \( (5, 4, -1) \). This process is essential as it gives a precise endpoint for the vector, confirming the movement described by the vector components.