Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the vector emanating from the origin whose terminal point is the midpoint of the segment joining \((3,2,-1)\) and \((5,-7,2)\).

Step-by-Step Solution

Answer

The vector is \((4, -2.5, 0.5)\).

Step 1: Identify the Midpoint Formula

To find the midpoint of a line segment defined by two points in three-dimensional space, use the formula:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]where \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are the coordinates of the two points.

Step 2: Substitute Given Points into Midpoint Formula

Substitute the coordinates of the points \((3,2,-1)\) and \((5,-7,2)\) into the midpoint formula:\[ M = \left( \frac{3+5}{2}, \frac{2+(-7)}{2}, \frac{-1+2}{2} \right) \].

Step 3: Calculate the Midpoint Coordinates

Perform the calculations for each component separately to find the midpoint:\[ M = \left( \frac{8}{2}, \frac{-5}{2}, \frac{1}{2} \right) \].Therefore, the midpoint is \((4, -2.5, 0.5)\).

Step 4: Determine the Vector from the Origin to the Midpoint

The vector emanating from the origin \((0,0,0)\) to the midpoint \((4, -2.5, 0.5)\) is simply \((4, -2.5, 0.5)\) because the origin doesn't alter the direction or magnitude of the vector.

Key Concepts

Midpoint Formula

The midpoint formula is an essential tool in geometry and vector calculus, especially when working with points in both two and three-dimensional spaces. It helps to identify the exact middle point of a line segment by averaging the coordinates of the endpoints. This formula is particularly useful in dividing the segment into two equal parts. In three-dimensional space, each coordinate must be calculated separately using:

• \( M_x = \frac{x_1 + x_2}{2} \)
• \( M_y = \frac{y_1 + y_2}{2} \)
• \( M_z = \frac{z_1 + z_2}{2} \)

Remember, this formula serves to find the average location of two points in space.
For example, if you're given points (3, 2, -1) and (5, -7, 2), apply the formula to each coordinate:

• \( M_x = \frac{3+5}{2} = 4 \)
• \( M_y = \frac{2+(-7)}{2} = -2.5 \)
• \( M_z = \frac{-1+2}{2} = 0.5 \)

Thus, the midpoint is (4, -2.5, 0.5), making it the vector from the origin to this computed midpoint.

Three-Dimensional Space

Three-dimensional space, often represented by the coordinates (x, y, z), is a mathematical construct that allows us to model the real world in a detailed manner. It gives depth along with height and width, crucial for describing spatial relationships and dynamics. In this space, points are defined by three numbers which denote distances from three perpendicular reference planes: the x, y, and z axes.

When calculating items such as lines, planes, or even spheres within this space, it's helpful to visualize or sketch them in a three-axis system. Each point, like (3, 2, -1) or (5, -7, 2) mentioned earlier, has a unique position that can be plotted in this 3D grid.

Understanding three-dimensional space requires interacting with these axes. Moving in this space involves considering how a point or object changes across all three coordinates, offering a comprehensive approach to solving geometric tasks. As such, vectors, like the one from the origin to (4, -2.5, 0.5), showcase both direction and magnitude in these extendable dimensions.

Line Segment

In vector calculus, a line segment is the straight path that connects two points, having a definite start and end. This concept is vital for understanding geometric relations and vector operations. In three-dimensional space, a line segment not only traverses horizontally or vertically but also adds depth, enhancing the complexity and richness of spatial relations.

The segment between two points, say (3, 2, -1) and (5, -7, 2), can be visualized as a short link or bridge connecting these coordinates in three-dimensional space.

• The concept of a midpoint allows us to evenly divide this line segment.
  
• It acts as a 'balance point,' making the segments on either side of it equal in length.

The understanding of line segments also aids in comprehending more complex geometric entities like polygons and polyhedra. Recognizing each line segment's role translates into a clearer grasp of larger structures, proving useful in both theoretical computations and practical applications like engineering and architecture.