Question

Graph Vector \(\overrightarrow{\mathbf{A}}\) has a magnitude of \(27 \mathrm{~m}\) and points in a direction \(32^{\circ}\) above the positive \(x\) axis. Vector \(\overrightarrow{\mathbf{B}}\) has a magnitude of \(35 \mathrm{~m}\) and points in a direction \(55^{\circ}\) below the positive \(x\) axis. Sketch the vectors \(\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}}\), and \(\overrightarrow{\mathbf{C}}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\). Estimate the magnitude and the direction angle of \(\overrightarrow{\mathbf{C}}\) from your sketch.

Short answer

Vector \(\overrightarrow{\mathbf{C}}\) has a magnitude of \(\approx 45.26 \text{ m}\) and is \(18.6^{\circ}\) below the positive x-axis.

Step-by-step solution

Break Down Vector Components

First, we need to find the component vectors for \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\). For \(\overrightarrow{\mathbf{A}}\), the magnitude is 27 m and it points 32° above the positive x-axis. To find the components, use:\[A_x = 27 \cos(32^{\circ}) \quad \text{and} \quad A_y = 27 \sin(32^{\circ})\]Similarly, for \(\overrightarrow{\mathbf{B}}\), which has a magnitude of 35 m and points 55° below the x-axis:\[B_x = 35 \cos(55^{\circ}) \quad \text{and} \quad B_y = -35 \sin(55^{\circ})\]

Calculate Vector Components

Calculate the components:- For \(\overrightarrow{\mathbf{A}}\): \[A_x = 27 \cos(32^{\circ}) \approx 22.88 \quad \text{and} \quad A_y = 27 \sin(32^{\circ}) \approx 14.31\]- For \(\overrightarrow{\mathbf{B}}\):\[B_x = 35 \cos(55^{\circ}) \approx 20.07 \quad \text{and} \quad B_y = -35 \sin(55^{\circ}) \approx -28.68\]

Find Components of Resultant Vector C

Add the components of \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) to find \(\overrightarrow{\mathbf{C}} = \overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}}\):\[C_x = A_x + B_x = 22.88 + 20.07 = 42.95\]\[C_y = A_y + B_y = 14.31 + (-28.68) = -14.37\]

Calculate Magnitude and Direction of Vector C

Now calculate the magnitude of \(\overrightarrow{\mathbf{C}}\):\[|\overrightarrow{\mathbf{C}}| = \sqrt{(C_x)^2 + (C_y)^2} = \sqrt{(42.95)^2 + (-14.37)^2} \approx 45.26 \text{ m}\]Calculate the direction of \(\overrightarrow{\mathbf{C}}\) using the arctangent function:\[\theta_C = \tan^{-1}\left(\frac{C_y}{C_x}\right) = \tan^{-1}\left(\frac{-14.37}{42.95}\right) \approx -18.6^{\circ}\]This angle is below the positive x-axis.

Sketch the Vectors

On a coordinate plane, draw vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) based on their respective components and angles. Then, draw vector \(\overrightarrow{\mathbf{C}}\) originating from the same point and extending according to the resultant vectors' magnitude and direction. Label all vectors with their calculated magnitudes and directions.

Key concepts

Vector Components

Understanding vector components is key when dealing with vector addition. A vector can be broken down into its horizontal and vertical parts, known as components. These components are particularly useful because they simplify calculations, especially when adding vectors. To find these components, you use trigonometric functions. For a vector with magnitude and angle given, the horizontal component (along the x-axis) can be found using the cosine function, while the vertical component (along the y-axis) uses the sine function. For instance, if vector \(\overrightarrow{\mathbf{A}}\) has a magnitude of 27 m, and it points 32° above the positive x-axis, its components would be:

• \(A_x = 27 \cos(32^{\circ})\)
• \(A_y = 27 \sin(32^{\circ})\)

Another vector, \(\overrightarrow{\mathbf{B}}\), with a magnitude of 35 m pointing 55° below the x-axis, will have components:

• \(B_x = 35 \cos(55^{\circ})\)
• \(B_y = -35 \sin(55^{\circ})\)

Here, the negative sign indicates the direction is downward.

Magnitude and Direction

Once the components are determined, you can easily find the magnitude and direction of any vector. The magnitude gives us the length of the vector and can be computed using the Pythagorean theorem:\[|\overrightarrow{\mathbf{C}}|= \sqrt{(C_x)^2 + (C_y)^2}\]This equation essentially combines the horizontal and vertical components into one single value, representing the vector's actual length when plotted.
Next, to determine the vector's direction, we typically use the arctangent function, which relates the ratio of the vector's vertical and horizontal components:\[\theta_C = \tan^{-1}\left(\frac{C_y}{C_x}\right)\]Remember, the calculated angle could be below or above the x-axis, depending on the signs of the components, which must be considered when interpreting the direction.

Resultant Vector

The resultant vector is what you get when you add two or more vectors together. This is often needed in physics and engineering to find the net effect of multiple force vectors on an object.
To find the resultant vector, simply add up the corresponding components from each vector:

• \(C_x = A_x + B_x = 22.88 + 20.07 = 42.95\)
• \(C_y = A_y + B_y = 14.31 + (-28.68) = -14.37\)

These sum both the horizontal and vertical effects, resulting in a new vector that represents the combined effect of \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\). The resultant vector \(\overrightarrow{\mathbf{C}}\) hence provides the total impact in one clear direction and magnitude.

Coordinate Plane Sketching

Visualizing vectors on a coordinate plane helps you understand their components and resultant easily. Start by drawing your x and y axes. When sketching a vector:

• Begin from the origin or a specified point.
• Mark the angle it makes with the positive x-axis.
• Draw the vector with a length proportional to its magnitude.

Place \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) according to these principles, using their respective angles and magnitudes.
Finally, from the same origin point, add the resultant vector \(\overrightarrow{\mathbf{C}}\). Ensure it aligns correctly with the components calculated for \(C_x\) and \(C_y\). Label each vector with its respective magnitude and direction, as this allows anyone interpreting the diagram to understand the vectors' interactions clearly. This visual method shows how vector addition combines various directional forces into a single clear outcome.