Problem 27 - College Physics

Question

Given two vectors, \(\overrightarrow{\mathrm{A}}\) which has a length of 10.0 and makes an angle of \(45^{\circ}\) below the \(-x\) -axis, and \(\overrightarrow{\mathbf{B}}\) which has an \(x\) -component of +2.0 and \(\mathrm{a} y\) -component of +4.0 (a) sketch the vectors on \(x-y\) axes, with all their "tails" starting at the origin, and (b) calculate \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\).

Step-by-Step Solution

Answer

Vector sum: (-5.07, -3.07)

Step 1: Sketching Vector A

Vector \(\overrightarrow{\mathbf{A}}\) is 10 units long and makes a \(45^{\circ}\) angle below the \(-x\)-axis. To accurately sketch it: Start from the origin, draw a line towards the \(-x\) direction, and then rotate it 45 degrees downward, forming a vector that points into the second quadrant.

Step 2: Sketching Vector B

Vector \(\overrightarrow{\mathbf{B}}\) has \(x\)- and \(y\)- components: +2 and +4, respectively. Start at the origin and move 2 units right (positive \(x\)-direction), then 4 units up (positive \(y\)-direction). Draw the resultant vector \(\overrightarrow{\mathbf{B}}\) ending at \((2, 4)\).

Step 3: Finding Components of Vector A

Since \(\overrightarrow{\mathbf{A}}\) is at an angle and has a known length and direction, we find its components using trigonometry. The \(x\)-component: \(A_x = - 10 \cos(45^{\circ}) = -7.07\). The \(y\)-component: \(A_y = - 10 \sin(45^{\circ}) = -7.07\). Both components are negative because vector A points to the left and downward.

Step 4: Adding Vector B to Vector A

To find \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\), add the corresponding components:\[(A_x + B_x, A_y + B_y) = (-7.07 + 2, -7.07 + 4) = (-5.07, -3.07).\]

Step 5: Resultant Vector

The resulting vector from the sum \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) has components \((-5.07, -3.07)\), pointing towards the third quadrant.

Key Concepts

Vector Components

When dealing with vectors, it’s important to understand that they can be broken down into parts known as vector components. This simplifies complex vector problems by handling each direction separately. For a vector in a two-dimensional plane, there are typically two components: the horizontal component (x-component) and the vertical component (y-component).

You can think of these components as shadows or projections of the original vector onto the axes of a coordinate system:

• The **x-component** is found using the cosine function: For a vector \(\overrightarrow{\mathrm{A}}\) of length \(L\), making an angle \(\theta\) with respect to the x-axis, the x-component is \(A_x = L \cos(\theta)\).
• The **y-component** uses the sine function: Here, it is \(A_y = L \sin(\theta)\).

For vector \(\overrightarrow{\mathbf{A}}\) in our exercise, with a length of 10 and a direction 45 degrees below the negative x-axis, its components are:

• \(A_x = -10 \cos(45^{\circ}) = -7.07\)
• \(A_y = -10 \sin(45^{\circ}) = -7.07\)

Understanding these component calculations allows us to add vectors more easily and it is fundamental in solving vector-related physics problems.

Trigonometry in Physics

Trigonometry is a crucial tool in physics, especially when it comes to analyzing vectors. It helps us describe how vectors behave spatially and to compute important characteristics like direction and magnitude using trigonometric functions like sine, cosine, and tangent.

In our exercise, vector \(\overrightarrow{\mathrm{A}}\) is analyzed using trigonometry. Here is how we can utilize trigonometric functions:

• **Angle Measurement:** Determine the direction of a vector with respect to a coordinate axis using angles. In our example, vector \(\overrightarrow{\mathrm{A}}\) forms a \(45^{\circ}\) angle below the x-axis.
• **Component Calculation:** Break down the vector into x and y components using sine and cosine as discussed earlier. The sign of these functions helps to identify the direction (negative when heading down or left).

By splitting a vector into components using trigonometry, we maintain the vector’s original properties while simplifying calculations. This method is essential in solving for vector addition, where components are summed in their respective directions.

Graphical Representation of Vectors

Physics problems often involve visualizing vectors on a graph as it provides a clear depiction of magnitude and direction.

In our example, sketching is an essential step:

• **Origin Point:** Always start drawing vectors from a common point, typically the origin for clarity.
• **Direction and Magnitude:** Draw each vector as an arrow, with length proportional to its magnitude, and precisely oriented according to given angles.
• **Vector \(\overrightarrow{\mathrm{A}}\):** Starts from the origin and points 45 degrees below the \(-x\)-axis, making it visible in the second quadrant.
• **Vector \(\overrightarrow{\mathrm{B}}\):** Has components, so it moves right to 2 units and up to 4 units, ending at point \((2, 4)\).

Graphically, the resultant vector of \(\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}}\) appears as a sum of these movements, adding their x and y components together to visually show direction and magnitude in a straightforward manner. Graphical representation aids in understanding the spatial orientation and interaction of vectors.