Question

Vector \(\overline{\mathbf{A}}\) points in the negative \(y\) direction and has a magnitude of \(5 \mathrm{~km}\). Vector \(\overrightarrow{\mathbf{B}}\) has a magnitude of \(15 \mathrm{~km}\) and points in the positive \(x\) direction. Use components to find the magnitude of (a) \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\), (b) \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\), and (c) \(\overrightarrow{\mathrm{B}}-\overrightarrow{\mathrm{A}}\).

Short answer

All three magnitudes are \(5\sqrt{10}\) km.

Step-by-step solution

Define Vectors in Component Form

Firstly, define both vectors \(\overline{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) in terms of their components. Given that \(\overline{\mathbf{A}}\) points in the negative \(y\) direction with a magnitude of \(5\, \text{km}\), it can be expressed as \(\overline{\mathbf{A}} = (0, -5)\, \text{km}\). For \(\overrightarrow{\mathbf{B}}\), which points in the positive \(x\) direction with a magnitude of \(15\, \text{km}\), it can be expressed as \(\overrightarrow{\mathbf{B}} = (15, 0)\, \text{km}\).

Calculate Components of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\)

Add the corresponding components of the vectors \(\overline{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) to find \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\). This is calculated as follows: \[(0 + 15, -5 + 0) = (15, -5)\, \text{km}\].

Find the Magnitude of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\)

To find the magnitude of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\), use the formula for the magnitude of a vector, \(\sqrt{x^2 + y^2}\). Applying it, we have: \[\sqrt{15^2 + (-5)^2} = \sqrt{225 + 25} = \sqrt{250} = 5\sqrt{10}\, \text{km}\].

Calculate Components of \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\)

Subtract the components of \(\overrightarrow{\mathbf{B}}\) from \(\overline{\mathbf{A}}\): \[(0 - 15, -5 - 0) = (-15, -5)\, \text{km}\].

Find the Magnitude of \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\)

Use the magnitude formula for the vector \((-15, -5)\): \[\sqrt{(-15)^2 + (-5)^2} = \sqrt{225 + 25} = \sqrt{250} = 5\sqrt{10}\, \text{km}\].

Calculate Components of \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}\)

Subtract the components of \(\overline{\mathbf{A}}\) from \(\overrightarrow{\mathbf{B}}\): \[(15 - 0, 0 - (-5)) = (15, 5)\, \text{km}\].

Find the Magnitude of \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}\)

For the vector \((15, 5)\), calculate the magnitude: \[\sqrt{15^2 + 5^2} = \sqrt{225 + 25} = \sqrt{250} = 5\sqrt{10}\, \text{km}\].

Key concepts

Vector Components

When dealing with vectors, understanding how to break them down into components is crucial. Vector components express the vector in terms of its horizontal (x-axis) and vertical (y-axis) directions. This simplification is extremely helpful, especially for calculations involving multiple vectors. Consider a vector aligned along a specific direction, like vector **A** in the original exercise, which is given as (0, -5) km:

• The first part, 0 km, represents no movement along the x-axis.
• The second part, -5 km, shows a movement of 5 km in the negative y direction.

Similarly, vector **B** is expressed as (15, 0) km, indicating it moves 15 km along the positive x direction, but doesn't move in the y direction. Using component form allows us to easily add or subtract these vectors by simply working with their corresponding components.

Magnitude Calculation

To find the magnitude of a vector, we use a formula derived from the Pythagorean theorem. A vector's magnitude is the "length" of the vector, which represents how far the vector carries a point from the origin. For a vector with components (x, y), the magnitude is calculated as follows: \[\sqrt{x^2 + y^2}\]Let's take the vector resulting from **A** + **B**: it's given as (15, -5) km. Plugging into the formula:

• Square the x component: \(15^2 = 225\).
• Square the y component: \((-5)^2 = 25\).
• Add these squares: \(225 + 25 = 250\).
• Finally, take the square root: \(\sqrt{250} = 5\sqrt{10}\), giving us the magnitude of the vector in km.

This method works consistently no matter how the vector is oriented.

Vector Subtraction

Vector subtraction can be understood as adding a vector in the opposite direction. When subtracting vectors, each component is treated individually. Consider vectors **A** and **B** again:For **A** - **B**, we find:

• Subtract the x components: \(0 - 15 = -15\)
• Subtract the y components: \(-5 - 0 = -5\)
• The result is \((-15, -5)\) km.

Similarly, for **B** - **A**, we calculate:

• Subtract the x components: \(15 - 0 = 15\)
• Subtract the y components: \(0 - (-5) = 5\)
• The result is \((15, 5)\) km.

Each subtraction operation reflects moving along the x-axis and y-axis oppositely, allowing us to determine new vector directions and magnitudes. Consider these steps carefully, as vector subtraction is a fundamental concept in physics and engineering applications.