Problem 19 - Thomas Calculus

Question

Express each vector in the form \(\mathbf{v}=v_{1} \mathbf{i}+\) \(v_{2} \mathbf{j}+v_{3} \mathbf{k}.\) \(\overrightarrow{A B}\) if \(A\) is the point \((-7,-8,1)\) and \(B\) is the point \((-10,8,1)\)

Step-by-Step Solution

Answer

\(\overrightarrow{AB} = -3\mathbf{i} + 16\mathbf{j}\).

Step 1: Identify the coordinates of points A and B

Point A has the coordinates \[ A = (-7, -8, 1) \]and point B has the coordinates \[ B = (-10, 8, 1) \].We will be using these coordinates to find the vector \(\overrightarrow{AB}\).

Step 2: Calculate the components of vector AB

The vector \(\overrightarrow{AB}\) is calculated by subtracting the coordinates of point A from the coordinates of point B.\[\overrightarrow{AB} = (B_x - A_x, B_y - A_y, B_z - A_z)\]Substituting the given points,\[\overrightarrow{AB} = (-10 - (-7), 8 - (-8), 1 - 1)\]

Step 3: Simplify the components

Let's perform the calculations for each component:- For the \(x\)-component: \[ -10 + 7 = -3 \]- For the \(y\)-component: \[ 8 + 8 = 16 \]- For the \(z\)-component: \[ 1 - 1 = 0 \]

Step 4: Write the vector in the required form

Now we can express the vector \(\overrightarrow{AB}\) as\[\overrightarrow{AB} = -3\mathbf{i} + 16\mathbf{j} + 0\mathbf{k}\].This is the vector in the form \(\mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k}\).

Key Concepts

Vector Subtraction

Vector subtraction is a fundamental operation in vector algebra. It involves finding the difference between two vectors. In the context of points like A and B, vector subtraction helps us determine the vector that points from A to B.

• To perform vector subtraction, subtract corresponding components of the vectors.
• The formula for the vector subtraction between vectors, where vector A's coordinates are \(x_1, y_1, z_1\) and B's coordinates are \(x_2, y_2, z_2\), is given by \((x_2 - x_1, y_2 - y_1, z_2 - z_1)\).
• This subtraction yields a new vector that represents the direction and distance from one point to another.

In our example, vector subtraction helped us find the vector \(\overrightarrow{AB}\) by subtracting each element of the coordinates from point A from point B. This process tells us precisely how vector A differs from vector B.

Vector Components

Vector components are the building blocks of vectors in two-dimensional or three-dimensional spaces. Each vector can be expressed as a combination of its components along the axes.

• In three-dimensional space, a vector \(\mathbf{v}\) is broken down into components along the \(x\), \(y\), and \(z\) axes, labeled typically as \(v_1, v_2, v_3\).
• These components identify how much traction the vector has in each direction.
• They are an essential way to express vectors because they simplify operations like addition or subtraction.

In our scenario, the vector \(\overrightarrow{AB} = -3\mathbf{i} + 16\mathbf{j} + 0\mathbf{k}\) clearly shows its x-component as -3, y-component as 16, and z-component as 0. This breakdown facilitates the understanding and manipulation of the vector.

Three-Dimensional Vectors

Three-dimensional vectors are pivotal in representing points, objects, or forces in space. Unlike two-dimensional vectors, they consider an additional plane, z, which translates to more complex and realistic modeling.

• A three-dimensional vector is denoted as \(\mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k}\).
• The \(\mathbf{i}, \mathbf{j}, \) and \(\mathbf{k}\) are unit vectors along the x, y, and z-axis, denoting direction for each component.
• This three-axis system allows vectors to represent such varied phenomena as wind speed, velocity of an airplane, or location in 3D space.

Using the vector \(\overrightarrow{AB}\) from our problem as a three-dimensional vector, we recognize how each component contributes to moving through space. The system we used to calculate and express it helps deepen comprehension and manage real-world physical situations that involve volume and depth.