Question

Two vectors, \(\overrightarrow{\mathbf{r}}\) and \(\overrightarrow{\mathbf{s}}\), lie in the \(x y\) plane. Their magnitudes are \(4.5\) and \(7.3\) units, respectively, whereas their directions are \(320^{\circ}\) and \(85^{\circ}\) measured counterclockwise from the positive \(x\) axis. What is the value of \(\overrightarrow{\mathbf{r}} \cdot \overrightarrow{\mathbf{s}}\) ?

Short answer

The value of \(\overrightarrow{r} \cdot \overrightarrow{s}\) is approximately -21.014 units.

Step-by-step solution

Calculate the angle between the vectors

The direction of \(\overrightarrow{r}\) is \(320^\circ\) and the direction of \(\overrightarrow{s}\) is \(85^\circ\). Since these angles are measured counterclockwise from the positive x-axis, the angle between the vectors (\(\theta\)) can be calculated as the absolute difference between the two angles. So, \(\theta = |320^\circ - 85^\circ| = 235^\circ\).

Convert the angle to radians

The cosine function in most calculators and programming languages uses radians instead of degrees, so convert \(\theta\) into radians: \(\theta_{\text{rad}} = 235^\circ * \(\pi / 180\) = 4.10152 \text{ rad}\).

Compute the dot product

The dot product of vectors \(\overrightarrow{r}\) and \(\overrightarrow{s}\) is defined as \(|\overrightarrow{r}| * |\overrightarrow{s}| * cos(\theta)\). Substituting the given magnitudes of the vectors and the calculated angle: \(\overrightarrow{r} \cdot \overrightarrow{s} = 4.5 * 7.3 * cos(4.10152) = -21.014 units\).

Key concepts

Dot Product Calculation

The dot product, also known as the scalar product, is a fundamental operation in vector algebra that measures the product of two vectors' magnitudes and the cosine of the angle between them. The formula is given as:
\[ \bz {A} \cdot \bz {B} = |\bz {A}| \times |\bz {B}| \times \cos(\theta) \]
Pictorially, if vector \(\bz {A}\) projects onto vector \(\bz {B}\), the dot product gives us the magnitude of this projection times the magnitude of \(\bz {B}\). In the context of our exercise, we're looking to calculate the dot product of vectors \(\bz r\) and \(\bz s\) with magnitudes of 4.5 and 7.3 units respectively, and an angle \(\theta\) of 235 degrees between them. After converting this angle to radians (which is necessary for most mathematical computations), we can apply the formula to find the dot product.

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance from a reference point known as the pole, similar to the way an angle and magnitude determine the position of a vector in the plane.
In polar coordinates, the angle is often represented by \(\theta\) and usually measured in degrees or radians from the positive x-axis, counterclockwise. The distance from the pole is analogous to the vector's magnitude. Our vectors \(\bz r\) and \(\bz s\) in the exercise have their directions given in degrees as measured from the positive x-axis, which corresponds to their polar angle, and their magnitudes correspond to the radial distance from the origin. Understanding polar coordinates is key to visualizing and solving problems related to vectors in a plane.

Converting Degrees to Radians

Degrees and radians are two units for measuring angles. There are 360 degrees or \(2\pi\) radians in a complete circle. To convert from degrees to radians, we use the conversion factor \(\pi/180\).
In the exercise, the angle between two vectors is 235 degrees, which we need to convert to radians to use in mathematical functions such as cosine.
The conversion is done as follows:
\[\theta_{\text{rad}} = \theta_{\text{deg}} \times \left(\frac{\pi}{180}\right)\]
Applying this to our angle:
\[235^\circ \times \left(\frac{\pi}{180}\right) = 4.10152 \text{ rad}\]
It's crucial for students to become comfortable with this conversion because radians are the standard unit of angular measure in most scientific and technical calculations.