Question

A vector \(\overrightarrow{\mathbf{A}}\) has a magnitude of \(40.0 \mathrm{~m}\) and points in a direction \(20.0^{\circ}\) below the positive \(x\) axis. A second vector, \(\overrightarrow{\mathbf{B}}\), has a magnitude of \(75.0 \mathrm{~m}\) and points in a direction \(50.0^{\circ}\) above the positive \(x\) axis. Sketch the vectors \(\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}}\), and \(\overrightarrow{\mathbf{C}}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\).

Short answer

Vectors are sketched with calculated components: A at (37.6, -13.7), B at (48.2, 57.5), and C as their sum at (85.8, 43.8).

Step-by-step solution

Understanding Vector \\hat{A} Direction

Vector \( \overrightarrow{\mathbf{A}} \) is 40.0 m long and is directed 20.0° below the positive x-axis. To calculate its components, we use the angles measured counterclockwise from the positive x-axis as reference. A vector 20° below the x-axis means it has a 340° counterclockwise direction.

Calculating Components of \\hat{A}

The x-component of \( \overrightarrow{\mathbf{A}} \) is \( A_x = A \cos \theta = 40.0 \cos(340^\circ) \). The y-component is \( A_y = A \sin \theta = 40.0 \sin(340^\circ) \). Evaluating these gives \( A_x \approx 37.6 \) and \( A_y \approx -13.7 \).

Understanding Vector \\hat{B} Direction

Vector \( \overrightarrow{\mathbf{B}} \) is 75.0 m long and directed 50.0° above the positive x-axis. This can directly be used as \( \theta = 50° \) from the x-axis.

Calculating Components of \\hat{B}

The x-component of \( \overrightarrow{\mathbf{B}} \) is \( B_x = B \cos \theta = 75.0 \cos(50^\circ) \). The y-component is \( B_y = B \sin \theta = 75.0 \sin(50^\circ) \). Evaluating these gives \( B_x \approx 48.2 \) and \( B_y \approx 57.5 \).

Finding \\hat{C} Components

Vector \( \overrightarrow{\mathbf{C}} = \overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}} \). To find its components, add corresponding components: \( C_x = A_x + B_x \approx 37.6 + 48.2 = 85.8 \), \( C_y = A_y + B_y \approx -13.7 + 57.5 = 43.8 \).

Sketch Vectors Diagrammatically

Draw the coordinate axes. 1. From the origin, draw \( \overrightarrow{\mathbf{A}} \) 37.6 units along x-axis and -13.7 units along y-axis (below x-axis).2. From the origin, draw \( \overrightarrow{\mathbf{B}} \) 48.2 units along x-axis and 57.5 units along y-axis (above x-axis).3. \( \overrightarrow{\mathbf{C}} \) should be drawn starting at the origin, 85.8 units along x-axis and 43.8 units along y-axis.

Key concepts

Vector Components

Understanding vector components is essential to break down a vector into its parts that align with the coordinate axes. This simplifies complex vector operations like addition or subtraction.

A vector can be visually represented as having both x and y components, which correspond to the horizontal and vertical directions on a plane. If a vector has a known magnitude and direction, we can use trigonometric functions to determine these components.

• The x-component of a vector is found using the formula: \( \,V_x = V \,\cos(\theta) \, \), where \( \,V\,\) is the vector's magnitude and \( \,\theta\,\) is the angle measured from the positive x-axis.
• The y-component is determined with \( \,V_y = V \,\sin(\theta) \, \).

For instance, with vector \( \,\overrightarrow{\mathbf{A}}\,\) having a magnitude of \(40.0 \,\mathrm{m}\) and an angle of \(340^{\circ}\,,\) its x-component is \(37.6 \,\mathrm{m}\) and y-component is \(-13.7 \,\mathrm{m}\). This breakdown allows us to manage vectors easily in equations, summing or subtracting them as needed.

Vector Diagram

A vector diagram is a graphical representation of one or more vectors. This visual aid helps in understanding the direction and relative magnitude of vectors in a coordinate system. When you sketch vectors, it's crucial to place them correctly based on their components.

Here’s how you can draw a clear vector diagram:

• Start by drawing the coordinate axes, with the horizontal line representing the x-axis and the vertical line the y-axis.
• For vector \(\overrightarrow{\mathbf{A}}\), plot 37.6 units along the positive x-axis and then 13.7 units downwards because it is below the x-axis. This gives the direction and magnitude proper to its angle.
• For vector \(\overrightarrow{\mathbf{B}}\), draw 48.2 units on the x-axis and 57.5 units upwards since it's 50° above it.
• Finally, trace vector \(\overrightarrow{\mathbf{C}}\) from the origin using the sum of the other vectors’ components: 85.8 units along x and 43.8 units on y.

This graphical approach aligns with the analytical method and reinforces the understanding of vector addition.

Trigonometry in Physics

Trigonometry is a branch of mathematics that deals with the relations of the angles and sides of triangles. It forms an essential tool in physics when dealing with vectors, which are directional quantities often described by angles.

In the context of vectors, trigonometry helps convert a vector’s magnitude and direction into its x and y components. This conversion uses:

• Cosine function for determining the adjacent side (x-component) in a right triangle.
• Sine function for the opposite side (y-component).

By applying these functions, we can accurately project a vector onto the coordinate system, which simplifies subsequent vector operations.

For example, to determine the components of vector \(\overrightarrow{\mathbf{A}}\), noting its direction of 20° below the positive x-axis as 340° (since we consider the angle counterclockwise from the positive x-axis). Using equations like \( A_x = 40.0 \cos(340°) \) and \( A_y = 40.0 \sin(340°) \), we ensure vector operations follow both direction and magnitude accurately.