Problem 38
Question
Find a. the direction of \(\vec{P}_{1} \vec{P}_{2}\) and \(\mathbf{b}\). the midpoint of line segment \(P_{1} P_{2}\). $$P_{1}(0,0,0) \quad P_{2}(2,-2,-2)$$
Step-by-Step Solution
Verified Answer
The direction is (2, -2, -2), and the midpoint is (1, -1, -1).
1Step 1: Understand the line segment direction
The direction of the line segment from a point \(P_1\) to another point \(P_2\) can be found using the vector \(\vec{P}_1\vec{P}_2\), which is computed as \(P_2 - P_1\). This vector represents the change in coordinates from \(P_1\) to \(P_2\).
2Step 2: Calculate the directional vector \(\vec{P}_1\vec{P}_2\)
Subtract the coordinates of point \(P_1(0,0,0)\) from the coordinates of point \(P_2(2,-2,-2)\) to find the vector:\[ \vec{P}_1\vec{P}_2 = (2 - 0, -2 - 0, -2 - 0) = (2, -2, -2) \].This vector \((2, -2, -2)\) gives the direction of \(\vec{P}_1\vec{P}_2\).
3Step 3: Understand the midpoint formula
The midpoint of a line segment with endpoints \(P_1(x_1, y_1, z_1)\) and \(P_2(x_2, y_2, z_2)\) can be found using the formula:\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \].This formula gives the average of the coordinates of the endpoints.
4Step 4: Calculate the midpoint of the line segment \(P_1P_2\)
Substitute the coordinates of \(P_1(0,0,0)\) and \(P_2(2,-2,-2)\) into the midpoint formula:\[ M = \left(\frac{0 + 2}{2}, \frac{0 + (-2)}{2}, \frac{0 + (-2)}{2}\right) = \left(1, -1, -1\right) \].The midpoint of the line segment \(P_1P_2\) is \((1, -1, -1)\).
Key Concepts
Directional VectorMidpoint FormulaLine Segment
Directional Vector
When dealing with vectors, understanding the direction is crucial. In vector calculus, the directional vector is a powerful tool. It helps determine the orientation of a line or a segment in space. To find the directional vector between two points, say \(P_1\) and \(P_2\), you subtract the coordinates of \(P_1\) from the coordinates of \(P_2\). This subtraction gives you a new vector \(\vec{P}_1\vec{P}_2\) that shows the direction from \(P_1\) to \(P_2\).
For example, given points \(P_1(0,0,0)\) and \(P_2(2,-2,-2)\), the directional vector is calculated as:
\[ \vec{P}_1\vec{P}_2 = (2 - 0, -2 - 0, -2 - 0) = (2, -2, -2) \]
This results in the vector \((2, -2, -2)\). This vector represents how far and in which direction you need to move from \(P_1\) to arrive at \(P_2\). So essentially, the components 2, -2, and -2 tell you to move 2 units in the x-direction, 2 units negatively in the y-direction, and 2 units negatively in the z-direction.
For example, given points \(P_1(0,0,0)\) and \(P_2(2,-2,-2)\), the directional vector is calculated as:
\[ \vec{P}_1\vec{P}_2 = (2 - 0, -2 - 0, -2 - 0) = (2, -2, -2) \]
This results in the vector \((2, -2, -2)\). This vector represents how far and in which direction you need to move from \(P_1\) to arrive at \(P_2\). So essentially, the components 2, -2, and -2 tell you to move 2 units in the x-direction, 2 units negatively in the y-direction, and 2 units negatively in the z-direction.
Midpoint Formula
Calculating the midpoint between two points in space gives you the point that is exactly halfway between them. This can be particularly useful in geometry and physics. The formula for the midpoint \(M\) of a line segment between two points \(P_1(x_1, y_1, z_1)\) and \(P_2(x_2, y_2, z_2)\) is:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \]
This formula effectively averages the x, y, and z coordinates of the two endpoints.
Using this formula, the midpoint of the segment with endpoints \(P_1(0,0,0)\) and \(P_2(2,-2,-2)\) can be calculated as:
\[ M = \left(\frac{0 + 2}{2}, \frac{0 - 2}{2}, \frac{0 - 2}{2}\right) = (1, -1, -1) \]
This means the midpoint of the segment is located at \( (1, -1, -1) \). This point neatly divides the segment into two equal parts.
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \]
This formula effectively averages the x, y, and z coordinates of the two endpoints.
Using this formula, the midpoint of the segment with endpoints \(P_1(0,0,0)\) and \(P_2(2,-2,-2)\) can be calculated as:
\[ M = \left(\frac{0 + 2}{2}, \frac{0 - 2}{2}, \frac{0 - 2}{2}\right) = (1, -1, -1) \]
This means the midpoint of the segment is located at \( (1, -1, -1) \). This point neatly divides the segment into two equal parts.
Line Segment
A line segment is a part of a line that is bounded by two distinct endpoints. Unlike a line, which extends infinitely, a line segment has a finite length.
In three-dimensional space, like in the example with points \(P_1(0,0,0)\) and \(P_2(2,-2,-2)\), the line segment connects these two points. The concept of a line segment is foundational in geometry and is used in various applications.
Here's a key thing to remember:
In three-dimensional space, like in the example with points \(P_1(0,0,0)\) and \(P_2(2,-2,-2)\), the line segment connects these two points. The concept of a line segment is foundational in geometry and is used in various applications.
Here's a key thing to remember:
- **Endpoints:** The points \(P_1\) and \(P_2\) are known as the endpoints of the segment. They decide where the segment starts and stops.
- **Length:** The length of a line segment can be calculated using the distance formula in 3D, which requires the difference between corresponding coordinates of the endpoints. However, this was not the focus here.
- **Midpoint and Directional Vector:** These are tools to extract more information from a line segment, like where its center is and where it leads.
Other exercises in this chapter
Problem 38
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