Problem 20 - Precalculus

Question

Assume that the coordinates of the points \(P\) \(Q, R, S,\) and \(O\) are as follows: \(P(-1,3) \quad Q(4,6) \quad R(4,3) \quad S(5,9) \quad O(0,0)\) Draw the indicated vector (using graph paper) and compute its magnitude. Compute the sums using the definition. Use the parallelogram law to compute the sums. $$\overrightarrow{S Q}+\overrightarrow{O R}$$

Step-by-Step Solution

Answer

The resultant vector is (3, 0) and its magnitude is 3.

Step 1: Calculate Vector Components

To find the vector components, subtract the coordinates of the tail from the head for each vector. For \( \overrightarrow{SQ} \), the head is \( Q(4,6) \) and the tail is \( S(5,9) \). Therefore, \( \overrightarrow{SQ} = (4-5, 6-9) = (-1, -3) \). For \( \overrightarrow{OR} \), the head is \( R(4,3) \) and the tail is \( O(0,0) \). Hence, \( \overrightarrow{OR} = (4-0, 3-0) = (4, 3) \).

Step 2: Calculate Sum of Vectors

Add the corresponding components of the vectors to find \( \overrightarrow{SQ} + \overrightarrow{OR} \). This gives: \[ (-1, -3) + (4, 3) = (-1+4, -3+3) = (3, 0) \].

Step 3: Compute the Magnitude of the Resultant Vector

The magnitude of a vector \((a, b)\) is given by \( \sqrt{a^2 + b^2} \). For vector \( (3, 0) \), its magnitude is \( \sqrt{3^2 + 0^2} = \sqrt{9} = 3 \).

Step 4: Application of the Parallelogram Law

The parallelogram law states that the sum of vectors \( \overrightarrow{u} \) and \( \overrightarrow{v} \) can be represented as the diagonal of a parallelogram with sides \( \overrightarrow{u} \) and \( \overrightarrow{v} \). This confirms our calculation of the vector sum \( \overrightarrow{SQ} + \overrightarrow{OR} = (3, 0) \), as the diagonal vector drawn would match this resultant vector.

Key Concepts

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a branch of mathematics that uses coordinates to represent and solve geometric problems. It forms the foundation for understanding locations and movements in a plane. In our exercise, we use coordinate geometry to determine specific points such as \( P(-1, 3) \), \( Q(4, 6) \), \( R(4, 3) \), \( S(5, 9) \), and \( O(0, 0) \).

Each point is labeled with coordinates that describe its position on the plane using an \((x, y)\) format. The first number in the pair, \(x\), is the horizontal distance from the origin, while the second number, \(y\), is the vertical distance. By describing points in this way, we can apply algebraic methods to geometric problems, making them easier to solve.

In vector addition problems, coordinate geometry helps us both visualize and compute vector components. It allows us to explore the direct paths between points as vectors. With each endpoint identified by coordinates, transitioning from descriptive geometry to precise calculations becomes straightforward and systematic.

Vector Components

Vector components are the pieces of a vector that define its position and direction along each axis of our coordinate system. To calculate these components, we subtract the coordinates of the vector's tail from those of its head. For instance, with \(\overrightarrow{SQ}\), the vector is defined from \( S(5, 9) \) to \(Q(4, 6)\). Thus, the vector components are \((4-5, 6-9) = (-1, -3)\).

Similarly, for \(\overrightarrow{OR}\), the vector starts at the origin \(O(0, 0)\) and ends at \(R(4, 3)\), giving vector components \((4-0, 3-0) = (4, 3)\).

These components are vital because they simplify the process of adding vectors, which involves adding the corresponding components of each vector. This approach breaks the problem into manageable parts, allowing us to deal with horizontal and vertical influences separately. Understanding vector components is key to solving problems involving vector addition and more complex vector operations.

Magnitude of Vectors

The magnitude of a vector is a measure of its length or size, indicating how far the vector extends from its starting point. It is calculated using the Pythagorean theorem, acknowledging the relationship between the vector's horizontal (\(x\)) and vertical (\(y\)) components. For a vector \((a, b)\), its magnitude is \( \sqrt{a^2 + b^2} \).

In our exercise, once the resultant vector \((3, 0)\) was determined from adding \(\overrightarrow{SQ} + \overrightarrow{OR}\), we then found its magnitude. This involved calculating \( \sqrt{3^2 + 0^2} = \sqrt{9} = 3 \).

This clear approach to finding vector magnitude is important, as it provides insight into the extent of displacement represented by the vector. By measuring magnitude, we gain a deeper understanding of vector length regardless of its direction within the coordinate plane.

Parallelogram Law

The parallelogram law is a geometric method used to determine the sum of two vectors. It states that if two vectors \(\overrightarrow{u}\) and \(\overrightarrow{v}\) are represented as adjacent sides of a parallelogram, then their sum \(\overrightarrow{u} + \overrightarrow{v}\) is described by the diagonal of the parallelogram originating from the same point.

In our problem, we used the components of the vectors \(\overrightarrow{SQ}\) and \(\overrightarrow{OR}\) to calculate their sum via component-wise addition, leading us to the resultant vector \((3, 0)\).

The parallelogram law confirms this result visually: if we imagine constructing a parallelogram with vectors \(\overrightarrow{SQ}\) and \(\overrightarrow{OR}\) as adjacent sides, the diagonal will indeed measure \((3, 0)\).

This law not only validates our computational method but also provides a visual representation of vector addition, illustrating how different vectors combine to create resultant vectors in a straightforward manner.