Problem 7 - Contemporary Precalculus

Question

Find a vector with the origin as initial point that is equivalent to the vector \(\overrightarrow{P Q}\). $$P=(-4,-8), Q=(-10,2)$$

Step-by-Step Solution

Answer

Answer: The vector equivalent to \(\overrightarrow{P Q}\) with the origin as the initial point is \(\overrightarrow{O'=(-6,10)}\).

Step 1: Find the coordinates of the vector \(\overrightarrow{P Q}\)

To find the coordinates of the vector \(\overrightarrow{P Q}\), we need to subtract the coordinates of the initial point \(P\) from the coordinates of the terminal point \(Q\). Using the given points \(P=(-4,-8)\) and \(Q=(-10,2)\), the coordinates of vector \(\overrightarrow{P Q}\) are: $$\overrightarrow{P Q} = Q - P = (-10 + 4, 2 + 8)=(-6,10)$$

Step 2: Translate the vector to the origin

Since the vector \(\overrightarrow{P Q}\) has the coordinates \((-6,10)\), the vector equivalent to \(\overrightarrow{P Q}\) with the origin as its initial point is simply: $$\overrightarrow{O'=(-6,10)}$$ So the vector with the origin as the initial point that is equivalent to the vector \(\overrightarrow{P Q}\) is \(\overrightarrow{O'=(-6,10)}\).

Key Concepts

Vector Translation

Imagine you have a point that needs to be moved, or translated, to a new position. In vector geometry, vector translation allows us to shift a vector to a specific location while maintaining its direction and magnitude intact.
In our exercise, we started with the vector \(\overrightarrow{P Q}\), which was positioned between points \(P\) and \(Q\). Our task was to translate this vector so that it starts at the origin point \((0,0)\). This process involves understanding that the vector maintains its properties, but its starting position changes.
To achieve this effect, you simply keep the calculated vector coordinates and assume the starting point is now the origin, without any changes to the direction or length of the vector. This concept is essential in various physics and engineering fields, where vectors are often re-positioned for analysis or computation.

Vector Coordinates

Vectors are defined by coordinates that indicate their direction and magnitude. Imagine them like arrows: they have a starting point and an endpoint. Vector coordinates allow us to precisely describe these points.
When calculating the coordinates of a vector \(\overrightarrow{P Q}\) in our exercise, we had to subtract the coordinates of the initial point \(P = (-4, -8)\) from the coordinates of the terminal point \(Q = (-10, 2)\). The result, \((-6,10)\), reflects the vector's path in a coordinate plane.
Understanding vector coordinates is like reading a map that guides us through space, indicating not only where something is but also how it moves from one place to another. They are essential for performing complex operations in several scientific and engineering tasks.

Subtraction of Points

Subtraction of points is a fundamental process in coordinate geometry used to find vectors, like \(\overrightarrow{P Q}\), from two points. This method determines the direction and distance between the points.
To find the vector \(\overrightarrow{P Q}\), subtracting the coordinates of \(P = (-4, -8)\) from \(Q = (-10, 2)\) was essential. This operation resulted in the vector \((-6,10)\).
Here is the step-by-step breakdown of this operation:

• Subtract the x-coordinate of \(P\) from the x-coordinate of \(Q\): \(-10 - (-4) = -10 + 4 = -6\).
• Subtract the y-coordinate of \(P\) from the y-coordinate of \(Q\): \(2 - (-8) = 2 + 8 = 10\).

These calculations provide the changes needed in the x and y directions, capturing the vector's essence.