Problem 39 - Thomas Calculus

Question

If \(\overrightarrow{A B}=\mathbf{i}+4 \mathbf{j}-2 \mathbf{k}\) and \(B\) is the point \((5,1,3),\) find \(A\)

Step-by-Step Solution

Answer

Point \(A\) is at \((4, -3, 5)\).

Step 1: Understand the vector equation

The vector \(\overrightarrow{AB}\) from point \(A\) to point \(B\) is given by \(\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}\). We also know the coordinates of point \(B\) are \((5,1,3)\). To find point \(A\), we will use the relationship: \(\overrightarrow{AB} = B - A\).

Step 2: Express \\(A\\) in terms of \\(B\\) and \\(\overrightarrow{AB}\\)

According to the vector formula, \(B - A = \overrightarrow{AB}\). This can be rearranged to find \(A\) as: \(A = B - \overrightarrow{AB}\).

Step 3: Substitute coordinates of \\(B\\) and \\(\overrightarrow{AB}\\) to find \\(A\\)

Substitute \(B = (5, 1, 3)\) and \(\overrightarrow{AB} = (1, 4, -2)\) into the equation: \(A = (5, 1, 3) - (1, 4, -2)\). Calculate each component separately:- For \(x\)-coordinate: \(5 - 1 = 4\)- For \(y\)-coordinate: \(1 - 4 = -3\)- For \(z\)-coordinate: \(3 - (-2) = 3 + 2 = 5\).

Step 4: Write the final coordinates of \\(A\\)

After performing the calculations, the coordinates of point \(A\) are \((4, -3, 5)\).

Key Concepts

Vector Addition

Vector addition is an essential concept in vector mathematics. It involves combining two or more vectors to attain a resultant vector. Each vector, in components, can be thought of as having magnitude and direction represented along specific axes.
When adding vectors, it is crucial to combine the same components.

• The x-components are added together.
• The y-components are added together.
• The z-components are added together.

This idea is fundamental when dealing with vectors in mathematical or physics problems. A practical feature of vector addition is that it follows the commutative property. This means \(\overrightarrow{A} + \overrightarrow{B} = \overrightarrow{B} + \overrightarrow{A}\).
Understanding vector addition allows us to derive the outcome of multiple vector quantities acting simultaneously on a point, as seen in our problem where we evaluate point coordinates using given vectors.

Coordinates

Coordinates play a crucial role in describing positions within a space. In three-dimensional space, each point is identified by an ordered triplet \( (x, y, z)\).
These values indicate the position of the point along the respective x, y, and z axes.
Understanding coordinates helps in visualizing vectors and performing operations like addition and subtraction. In the problem above, we used coordinates from point B, \( (5, 1, 3) \), to determine the unknown coordinates of point A.
By breaking down components into separate calculations like x, y, and z values, we gain a clear understanding of how transformation and localization occur in 3D space.

Vector Subtraction

Vector subtraction is the process of finding a vector difference between two vectors, often representing navigation from one point to another.
To subtract vectors, simply subtract corresponding components one at a time:

• Subtract the x-components from each other.
• Subtract the y-components from each other.
• Subtract the z-components from each other.

The result is \(\overrightarrow{A} - \overrightarrow{B}\), which represents a vector pointing from the terminal point of \(\overrightarrow{B}\) to the terminal point of \(\overrightarrow{A}\). This is precisely used in our solution to find point A by rearranging the formula \( B - A = \overrightarrow{AB} \) to solve for A, demonstrating how vector subtraction can determine points' relative positions.

Three-Dimensional Vectors

Three-dimensional vectors are critical in representing motion and distance in space. They have three components: typically denoted by x, y, and z.
The essence of a 3D vector is its ability to capture a linear trajectory or position in a space with length, width, and depth.
Vectors in three dimensions can represent transformations and transitions effectively.

• For example, the vector \( \overrightarrow{AB} \) is used to express the direction and magnitude of the path from A to B in our example.
• Three-dimensional vectors help in calculating real-world spatial problems, given their capacity to encompass different dimensional shifts.

In equations, like those within our solution process, they facilitate operations that allow us to isolate positions, demonstrating the power of vectors in spatial calculations.