Problem 19 - University Calculus: Early Transcendentals

Question

Express each vector in the form \(\mathbf{w}=w_{1} \mathbf{i}+\) \(w_{2} \mathbf{j}+w_{3} \mathbf{k}\). \(\overrightarrow{A B}\) 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: Identifying Coordinates

Begin by defining the coordinates for points A and B. Point A has the coordinates \((-7, -8, 1)\) and point B has the coordinates \((-10, 8, 1)\). These coordinates will help us find the vector from A to B.

Step 2: Calculate the Vector Components

To find the components of vector \(\overrightarrow{AB}\), we use the formula: \[ \overrightarrow{AB} = (B_x - A_x) \mathbf{i} + (B_y - A_y) \mathbf{j} + (B_z - A_z) \mathbf{k} \] Substitute the given coordinates: \[ \overrightarrow{AB} = (-10 - (-7)) \mathbf{i} + (8 - (-8)) \mathbf{j} + (1 - 1) \mathbf{k} \].

Step 3: Simplify the Components

Calculate each component of the vector: - For \(\mathbf{i}\): \[ w_1 = -10 + 7 = -3 \] - For \(\mathbf{j}\): \[ w_2 = 8 + 8 = 16 \] - For \(\mathbf{k}\): \[ w_3 = 1 - 1 = 0 \].

Step 4: Express the Vector in Component Form

Now that we have the components, express the vector \(\overrightarrow{AB}\) using \(\mathbf{i}, \mathbf{j}, \mathbf{k}\): \[ \overrightarrow{AB} = -3 \mathbf{i} + 16 \mathbf{j} + 0 \mathbf{k} \].

Key Concepts

Vector Components

Vectors are essential elements in physics and mathematics, helping us describe quantities with both magnitude and direction. To fully express a vector, we use its components along specific directions or axes. For example, in three-dimensional space, we often use

• x-direction, represented by the unit vector \(\mathbf{i}\)
• y-direction, represented by the unit vector \(\mathbf{j}\)
• z-direction, represented by the unit vector \(\mathbf{k}\).

Breaking a vector into these components makes calculations involving vectors much more manageable. The component along each axis tells us how much of the vector acts in that specific direction.

In the exercise, we calculated the vector \(\overrightarrow{AB}\) by finding differences in the x, y, and z coordinates. This involves subtracting the coordinates of points A and B:

• The x-component: \(B_x - A_x\) gives the presence along \(\mathbf{i}\).
• The y-component: \(B_y - A_y\) reveals the vector's action along \(\mathbf{j}\).
• The z-component: \(B_z - A_z\) indicates its effect in the \(\mathbf{k}\) direction.

Understanding vector components allows us to translate our multi-dimensional problems into more straightforward one-dimensional pieces that are easier to handle.

Coordinate Geometry

Coordinate geometry is a fundamental branch of mathematics that combines geometry and algebra to analyze figures using coordinates on a plane or in space.
It links the geometric representation of shapes with the algebraic equations and expressions that describe them.

In the context of our vector exercise, we used the concepts of coordinate geometry to describe both the points and the vector:

• Points as coordinates: We expressed point A as \((-7, -8, 1)\) and point B as \((-10, 8, 1)\). Each set of numbers represents a specific position on the coordinate plane.

The use of coordinates helps us accurately perform calculations to determine vectors or any other geometric entities. Here, the coordinates helped identify the path from point A to B, crucial for determining the vector \(\overrightarrow{AB}\).
Coordinate geometry is not just limited to vectors; it is also vital in studying lines, curves, surfaces, and different shapes in space using algebraic methods.

Unit Vectors

Unit vectors are special vectors with a magnitude of exactly one. They serve as building blocks in vector calculation, representing directions on a coordinate plane without any specific quantity but purely as a directional indicator.

Think of unit vectors as the compass directions of vectors. In three-dimensional space, the unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) align with the x, y, and z-axes, respectively:

• \(\mathbf{i}\) points along the positive x-axis.
• \(\mathbf{j}\) aligns with the positive y-axis.
• \(\mathbf{k}\) points in the direction of the positive z-axis.

In our exercise, each component of the vector \(\overrightarrow{AB}\) was multiplied by these unit vectors to properly express it in component form. As such, unit vectors help maintain clarity and directionality in vector expressions.
Moreover, any vector can be expressed as a combination of unit vectors multiplied by scalar values, which are the magnitudes along the respective axes. This simplification is crucial, making complex vector operations much more approachable.