Problem 5 - Calculus for Biology and Medicine

Question

Let \(A=(0,1,-3)\) and \(B=(-1,1,2)\). Find the vector representation of \(\overrightarrow{A B}\).

Step-by-Step Solution

Answer

The vector \(\overrightarrow{AB}\) is \((-1, 0, 5)\).

Step 1: Understand the Vector Representation

The vector \(\overrightarrow{AB}\) represents the direction and magnitude of the line from point \(A\) to point \(B\). It is found by subtracting the coordinates of point \(A\) from the coordinates of point \(B\). This is based on the geometric interpretation of vectors from one point to another.

Step 2: Subtract the Coordinates

To find the vector \(\overrightarrow{AB}\), subtract the coordinates of \(A\) from \(B\), using the formula: \(\overrightarrow{AB} = (x_2-x_1, y_2-y_1, z_2-z_1)\).For this exercise, the points \(A=(0,1,-3)\) and \(B=(-1,1,2)\):- The \(x\)-coordinate: \(-1 - 0 = -1\)- The \(y\)-coordinate: \(1 - 1 = 0\)- The \(z\)-coordinate: \(2 - (-3) = 2 + 3 = 5\)

Step 3: Write the Vector Representation

After calculating the differences, write the vector \(\overrightarrow{AB}\) using the components found:\[\overrightarrow{AB} = (-1, 0, 5)\]

Key Concepts

Coordinate Subtraction

Coordinate subtraction is a method used to find the vector between two points in space. To better understand this, consider two points, \(A = (x_1, y_1, z_1)\) and \(B = (x_2, y_2, z_2)\). To find the vector \(\overrightarrow{AB}\), you calculate the difference in each coordinate from \(A\) to \(B\).
This process involves subtracting the corresponding coordinates of point \(A\) from point \(B\):

• \(x\)-coordinate: \(x_2 - x_1\)
• \(y\)-coordinate: \(y_2 - y_1\)
• \(z\)-coordinate: \(z_2 - z_1\)

This subtraction gives the components of the vector, which specifies its position in the space.
A practical example using the points \(A=(0,1,-3)\) and \(B=(-1,1,2)\) involves the following steps:

• Subtract the \(x\)-coordinates: \(-1 - 0 = -1\)
• Subtract the \(y\)-coordinates: \(1 - 1 = 0\)
• Subtract the \(z\)-coordinates: \(2 - (-3) = 5\)

Thus, the vector representation of \(\overrightarrow{AB}\) is \((-1, 0, 5)\).

Direction and Magnitude

The vector \(\overrightarrow{AB}\) not only shows the difference in position between points \(A\) and \(B\), but also conveys both direction and magnitude.
Direction in this context means which way the vector is pointing from \(A\) to \(B\). The components \((-1, 0, 5)\) help us visualize this:

• The \(x\)-component \(-1\) indicates movement in the negative x-direction.
• The \(y\)-component \(0\) indicates no movement in the y-direction.
• The \(z\)-component \(5\) indicates movement in the positive z-direction.

Magnitude, on the other hand, is the length of the vector.
It can be calculated using the Pythagorean theorem extended to three dimensions:
\[\|\overrightarrow{AB}\| = \sqrt{(-1)^2 + 0^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26}\]
This length, or magnitude, tells how far apart the points are in space.

Geometric Interpretation

Geometrically, vectors and their representation provide significant insights into spatial relationships between points.
In the example of the vector \(\overrightarrow{AB}\), the points \(A = (0,1,-3)\) and \(B = (-1,1,2)\) can be visualized in a three-dimensional space and connected by the vector \((-1, 0, 5)\).
Imagine a straight line starting from point \(A\) and ending at point \(B\). The vector \(\overrightarrow{AB}\) embodies this line and illustrates not just its path, but also its orientation and size.
This visualization aids in understanding vectors as arrows with definite positions, angles, and lengths.
Each component of the vector affects its overall shape and direction in 3D space. For example:

• If all components are positive, the vector moves diagonally upwards.
• If some are negative, it shifts the whole vector in the opposite direction along those axes.

By breaking vectors down into their components, we get clear insights into their function and importance in geometry and physics.