Problem 16 - 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. $$\overrightarrow{O S}+\overrightarrow{Q O}$$

Step-by-Step Solution

Answer

The vector sum is \( \langle 1, 3 \rangle \) with a magnitude of \( \sqrt{10} \).

Step 1: Initialize the vectors

First, identify the vectors given by the points. Vector \( \overrightarrow{OS} \) starts at the origin \( O(0,0) \) and ends at point \( S(5,9) \). Therefore, \( \overrightarrow{OS} = \langle 5, 9 \rangle \). Vector \( \overrightarrow{QO} \) starts at point \( Q(4,6) \) and ends at the origin \( O(0,0) \), thus \( \overrightarrow{QO} = \langle 0-4, 0-6 \rangle = \langle -4, -6 \rangle \).

Step 2: Compute the vector sum using components

Add corresponding components of the vectors to compute \( \overrightarrow{OS} + \overrightarrow{QO} \). \[ \overrightarrow{OS} + \overrightarrow{QO} = \langle 5, 9 \rangle + \langle -4, -6 \rangle = \langle 5 - 4, 9 - 6 \rangle = \langle 1, 3 \rangle \]

Step 3: Compute the magnitude of the resulting vector

Use the formula for the magnitude of a vector \( \langle a, b \rangle \) which is given by \( \sqrt{a^2 + b^2} \). For vector \( \langle 1, 3 \rangle \), calculate its magnitude as follows: \[ \text{Magnitude} = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \].

Step 4: Apply the Parallelogram Law

To apply the parallelogram law, imagine vectors \( \overrightarrow{OS} \) and \( \overrightarrow{QO} \) as adjacent sides of a parallelogram. The diagonal of the parallelogram represents the vector sum \( \overrightarrow{OS} + \overrightarrow{QO} \), which we computed to be \( \langle 1, 3 \rangle \). The magnitude \( \sqrt{10} \) remains the same as previously calculated.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows us to find the exact position of points, lines, and various shapes using coordinates, typically in the form \((x, y)\). In the original exercise, each point, such as \(P(-1,3)\), \(Q(4,6)\), or \(O(0,0)\), is defined by a pair of numbers that tell us where to find this point on the Cartesian plane.

The coordinate system makes it easier to calculate distances between points and perform operations like vector addition. By using coordinates, we can transform geometric problems into algebraic equations, which are often simpler to solve.

• Points on the plane are often denoted by letters, such as \(P\), \(Q\), \(O\).
• Each point has an \(x\)-coordinate and a \(y\)-coordinate.
• Vectors can be expressed as differences between coordinates.

This translation from geometric to algebraic terms is what makes coordinate geometry such a powerful tool in problem-solving.

Vector Magnitude

The magnitude of a vector is a value that represents its length or size. In coordinate geometry, vectors are often described using their components, such as \(\langle a, b \rangle\). The magnitude gives us a scalar measure of how long the vector is, irrespective of its direction.

To calculate the magnitude of a vector \( \langle a, b \rangle \), use the Pythagorean Theorem: \[\text{Magnitude} = \sqrt{a^2 + b^2}\]

• This formula is derived from the distance formula in coordinate geometry.
• For the vector \( \langle 1, 3 \rangle \) in the exercise, this results in a magnitude of \( \sqrt{10} \).
• The units of magnitude are the same as those used for the vector's components.

The magnitude tells us the 'size' of the vector but does not give us any information about its direction.

Parallelogram Law

The parallelogram law is a visual method used to find the sum of two vectors. According to this law, if two vectors are represented as adjacent sides of a parallelogram, their resultant, or sum, is represented by the diagonal of the parallelogram starting from the point where the vectors originate.

When applying this law:

• Draw both vectors starting from the same point.
• Complete the shape of the parallelogram using these vectors as sides.
• The vector sum is the diagonal of the parallelogram.

In the exercise, the vectors \(\overrightarrow{OS}\) and \(\overrightarrow{QO}\) are regarded as the sides of the parallelogram. Their sum, \(\langle 1, 3\rangle\), is shown as the diagonal. This resulting vector tells us both the direction and magnitude, effectively condensing the influences of both original vectors into one.