Problem 12 - Thomas Calculus

Question

Find the component form of the vector. The sum of \(\overrightarrow{A B}\) and \(\overrightarrow{C D},\) where \(A=(1,-1), B=(2,0),\) \(C=(-1,3),\) and \(D=(-2,2)\)

Step-by-Step Solution

Answer

The component form of the vector sum is \((0, 0)\).

Step 1: Calculate Vector AB

To find the vector \(\overrightarrow{AB}\), subtract the coordinates of point \(A\) from point \(B\). The formula is: \(\overrightarrow{AB} = (B_x - A_x, B_y - A_y)\). Therefore, \(\overrightarrow{AB} = (2 - 1, 0 - (-1)) = (1, 1)\).

Step 2: Calculate Vector CD

To find the vector \(\overrightarrow{CD}\), subtract the coordinates of point \(C\) from point \(D\) using the formula: \(\overrightarrow{CD} = (D_x - C_x, D_y - C_y)\). Thus, \(\overrightarrow{CD} = (-2 - (-1), 2 - 3) = (-1, -1)\).

Step 3: Add Vectors AB and CD

To find the sum of vectors \(\overrightarrow{AB}\) and \(\overrightarrow{CD}\), add their respective components using: \(\overrightarrow{AB} + \overrightarrow{CD} = (AB_x + CD_x, AB_y + CD_y)\). So, \(\overrightarrow{AB} + \overrightarrow{CD} = (1 + (-1), 1 + (-1)) = (1 - 1, 1 - 1) = (0, 0)\).

Key Concepts

Vector Components

When we talk about vectors, we often break them down into simpler parts called components. Imagine a vector as an arrow. Instead of thinking about where the entire arrow points, we can describe it using the lengths of its shadow on the x-axis and y-axis. These are known as the vector components. Using coordinate points, vector components can be represented as differences:

• The x-component: calculated as the difference between the x-coordinates of the two points.
• The y-component: calculated as the difference between the y-coordinates.

For example, vector \( \overrightarrow{AB} \) from point \( A \) to point \( B \) is found by subtracting the coordinates: \( B_x - A_x \) for the x-component, and \( B_y - A_y \) for the y-component. So for \( A = (1, -1) \) and \( B = (2, 0) \), the components are \( (1, 1) \). This means the vector can be represented by moving 1 unit right on the x-axis and 1 unit up on the y-axis.
Breaking vectors into components is useful because it simplifies calculations, like adding vectors together, which involves just adding their respective components.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is where algebra meets geometry on a Cartesian plane. Each point here is defined using an ordered pair of numbers,

• the x-coordinate indicating how far along the horizontal axis the point is,
• and the y-coordinate showing how far along the vertical axis the point is.

In the context of vectors, coordinate geometry helps us visualize the position and movement from one point to another. For instance, when calculating vector \( \overrightarrow{CD} \) from points \( C \) and \( D \), the coordinates of these points guide us: \

• Starting at \( C = (-1, 3) \),\
• ending at \( D = (-2, 2) \)

provide a difference of \( (-1, -1) \) in vector form, illustrating a movement left and down on the plane. Coordinate geometry not only allows us to find these movements mathematically but also provides a means to plot and visualize them, making these operations intuitive and accessible.

Vector Subtraction

Vector subtraction is similar to vector addition but involves taking away components instead of adding them. Imagine starting at the tail of one vector and aiming to end at the tail of another. To find the vector subtraction pattern, you subtract the components of one vector from the other. In our problem,

• to find \( \overrightarrow{AB} \), we subtracted the coordinates of \( A \) from \( B \), resulting in \( (1, 1) \).
• This subtraction operation is performed component-wise.

Similarly, subtracting the coordinates of \( C \) from \( D \) gives vector \( \overrightarrow{CD} = (-1, -1) \). The subtraction results in vectors that tell us precisely how much we need to "move" from one point to another on the plane. Performing the subtraction correctly ensures that vector tasks like finding positions or calculating distances are carried out accurately. Through vector subtraction, we can derive useful information like direction and quantify absolute differences between points in coordinate geometry.