Problem 26 - 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. $$\overrightarrow{S Q}+\overrightarrow{S R}$$

Step-by-Step Solution

Answer

The sum vector is \((-2, -9)\), and its magnitude is \(\sqrt{85}\).

Step 1: Identify Vector Components

The vector \(\overrightarrow{SQ}\) starts at point \(S(5, 9)\) and ends at point \(Q(4, 6)\). The components are calculated by subtracting the coordinates of \(S\) from \(Q\):\[\overrightarrow{SQ} = (4 - 5, 6 - 9) = (-1, -3)\]Similarly, the vector \(\overrightarrow{SR}\) starts at point \(S(5, 9)\) and ends at \(R(4, 3)\):\[\overrightarrow{SR} = (4 - 5, 3 - 9) = (-1, -6)\]

Step 2: Compute Vector Sum

The sum of the vectors \(\overrightarrow{SQ}\) and \(\overrightarrow{SR}\) can be found by adding their components:\[\overrightarrow{SQ} + \overrightarrow{SR} = (-1, -3) + (-1, -6) = (-1 + -1, -3 + -6) = (-2, -9)\]

Step 3: Compute Magnitude of the Resultant Vector

The magnitude of a vector \((a, b)\) is calculated as:\[\|\overrightarrow{a}\| = \sqrt{a^2 + b^2}\]For the vector \((-2, -9)\), the magnitude is:\[\|\overrightarrow{-2, -9}\| = \sqrt{(-2)^2 + (-9)^2} = \sqrt{4 + 81} = \sqrt{85}\]

Step 4: Apply the Parallelogram Law (Conceptual)

According to the parallelogram law, if two vectors originate from the same point, the resultant vector is along the diagonal of the parallelogram formed. The coordinates and resultant magnitude remain consistent with our calculations.

Key Concepts

Parallelogram Law

When you have two vectors that originate from the same point, you can use the parallelogram law to find their resultant vector. Imagine drawing both vectors such that their tails are at the same starting point. The length and direction formed by the diagonal of the parallelogram they create give you the resultant vector.
This law not only helps in visualizing vector addition but also ensures that the mathematical calculations are accurate.

• To apply it, place the vectors tail-to-tail.
• Draw a parallelogram with the two vectors as adjacent sides.
• The diagonal of the parallelogram represents the vector sum.

In simple cases, you can quickly visualize this with graph paper, making it an intuitive way to understand vector addition.

Magnitude of a Vector

The magnitude of a vector gives you an idea of its length or size without concerning its direction. When dealing with a two-dimensional vector like \((a, b)\), its magnitude is found using the Pythagorean theorem.
The calculation involves squaring both components, adding them, and then taking the square root of the result. This straightforward process gives you a single number representing how long the vector is.

• For a vector \(\overrightarrow{v} = (a, b)\), the formula is \[ \|\overrightarrow{v}\| = \sqrt{a^2 + b^2} \]
• This magnitude represents distance in a plane and is always a non-negative value.

Understanding the magnitude is essential in various applications, such as physics problems involving forces, velocities, and more.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to represent geometric figures using a coordinate system. With this approach, points are defined by coordinates, commonly in the form \(x, y\) in a 2D plane.
In vector problems, coordinates help in determining vector components, which are crucial for further calculations like vector addition, subtraction, and finding magnitudes.

• Each vector can be broken down into components along the x and y axes using its start and end coordinates.
• This approach simplifies computing complex vector operations by transforming them into straightforward arithmetic tasks.

Utilizing coordinate geometry, one can easily visualize geometric shapes and perform algebraic operations to solve problems efficiently.