Question

Equality of vectors Assume \(\overrightarrow{P Q}\) equals \(\overrightarrow{R S}\). Does it follow that \(\overrightarrow{P R}\) is equal to \(\overrightarrow{Q S} ?\) Explain your answer.

Short answer

Answer: Yes, if \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) are equal, then \(\overrightarrow{P R}\) and \(\overrightarrow{Q S}\) are also equal.

Step-by-step solution

Understand the given vectors equality

We are given that the vectors \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) are equal. This means that their magnitudes and directions are the same.

Establish the relationship between the vectors components

Let \(\overrightarrow{P Q} = \begin{pmatrix} p_1 - q_1 \\ p_2 - q_2 \end{pmatrix}\) and \(\overrightarrow{R S} = \begin{pmatrix} r_1 - s_1 \\ r_2 - s_2 \end{pmatrix}\). Given that \(\overrightarrow{P Q} = \overrightarrow{R S}\), their components must be equal which gives us the following equations: 1. \(p_1 - q_1 = r_1 - s_1\) 2. \(p_2 - q_2 = r_2 - s_2\)

Determine whether \(\overrightarrow{P R}\) and \(\overrightarrow{Q S}\) are equal

Now, let's find \(\overrightarrow{P R}\) and \(\overrightarrow{Q S}\): \(\overrightarrow{P R} = \begin{pmatrix} p_1 - r_1 \\ p_2 - r_2 \end{pmatrix}\) \(\overrightarrow{Q S} = \begin{pmatrix} q_1 - s_1 \\ q_2 - s_2 \end{pmatrix}\) For these vectors to be equal, the following conditions must hold: 1. \(p_1 - r_1 = q_1 - s_1\) 2. \(p_2 - r_2 = q_2 - s_2\) Now we have two sets of simultaneous equations. Let's compare the equations in each set: From equations 1 and 2 in Step 2, and using the transitive property, we have: \(p_1 - q_1 = r_1 - s_1 \Rightarrow p_1 - r_1 = q_1 - s_1\) \(p_2 - q_2 = r_2 - s_2 \Rightarrow p_2 - r_2 = q_2 - s_2\) These equations are the same as the conditions we derived in Step 3 for \(\overrightarrow{P R}\) and \(\overrightarrow{Q S}\) to be equal. Hence, as \(\overrightarrow{P Q} = \overrightarrow{R S}\), \(\overrightarrow{P R}\) must equal \(\overrightarrow{Q S}\).

Key concepts

Vector Components

When discussing vectors, an essential aspect is their components. These components help us understand how a vector is expressed in terms of its basic directional parts. For two-dimensional vectors, we represent it as

• A horizontal component, denoted as the first coordinate (usually marked as \(x\) or \(p_1\)),
• A vertical component, marked as the second coordinate (often labeled as \(y\) or \(p_2\)).

Vectors like \(\overrightarrow{PQ}\) can be expressed with the components \(p_1 - q_1\) for the x-direction, and \(p_2 - q_2\) for the y-direction. Knowing the vector components gives us precise detail of each vector's direction and magnitude.
By comparing the components of vectors \(\overrightarrow{PQ}\) and \(\overrightarrow{RS}\), where equality of vector components indicates that the two vectors are equal, you establish a clear understanding of their nature.
Learning to translate vector equality into equations like \(p_1 - q_1 = r_1 - s_1\) is a fundamental skill for solving problems related to vector equality.

Magnitude and Direction

Magnitude and direction are two critical attributes describing a vector fully. The magnitude is essentially the "size" or "length" of the vector, calculated using the Pythagorean theorem for two-dimensional vectors. For a vector like \(\overrightarrow{PQ} = \begin{pmatrix} p_1 - q_1 \ p_2 - q_2 \end{pmatrix}\), the magnitude \(|\overrightarrow{PQ}|\) is given as

• \( |\overrightarrow{PQ}| = \sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2}\).

The direction relates to the angle this vector makes with a reference axis, providing the tilt or orientation of the line it forms.
When two vectors possess the same magnitude and direction, they are identical by definition. Hence, the given vectors \(\overrightarrow{PQ}\) and \(\overrightarrow{RS}\) being equal means they have not only the same length but also point in the same direction. This property is crucial when identifying if any two vectors are indeed equivalent.
If a vector's components are equal, then by default, its magnitude and direction must align as well, leading to vector equality.

Simultaneous Equations

Simultaneous equations are a powerful tool that can be used to solve problems involving vector components. They come into play when multiple equations need to be solved together, offering a method to find values that satisfy all conditions at once.
In vector equality problems, like comparing \(\overrightarrow{PR}\) and \(\overrightarrow{QS}\), simultaneous equations help verify equality. We derive these equations from vector components' equality:

• First equation: \(p_1 - r_1 = q_1 - s_1\),
• Second equation: \(p_2 - r_2 = q_2 - s_2\).

These equations represent the idea that both the horizontal and vertical displacements must match for the vectors to be equal.
Solving these simultaneous equations confirms the relationships between different points, such as proving that \(\overrightarrow{PR}\) equals \(\overrightarrow{QS}\) when \(\overrightarrow{PQ}\) and \(\overrightarrow{RS}\) are equal.
Understanding simultaneous equations allows students to isolate variables and solve diverse mathematical challenges, further reinforcing their grasp of vector equality.