Problem 29 - Contemporary Precalculus

Question

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=10, \theta=225^{\circ}$$

Step-by-Step Solution

Answer

Question: Given the magnitude of a vector, 10 units, and its direction angle, 225 degrees, find the vector in component form. Answer: The vector in component form is approximately <-7.071, -7.071>.

Step 1: Convert direction angle to radians

The direction angle is given in degrees, so we must convert it to radians first. To convert from degrees to radians, we can use the formula: $$\text{radians} = \frac{\text{degrees} \times \pi}{180}$$ In our case, we have: $$\theta=225^{\circ} \times \frac{\pi}{180} = \frac{5\pi}{4} \, \text{radians}$$

Step 2: Use polar-to-cartesian conversion formulas

Now that we have the magnitude and direction angle in radians, we can find the components of the vector using the following polar-to-cartesian conversion formulas: $$x = \| \mathbf{v} \| \cos{\theta}$$ $$y = \| \mathbf{v} \| \sin{\theta}$$ Using our values, we get: $$x = 10 \cos{(\frac{5\pi}{4})}$$ $$y = 10 \sin{(\frac{5\pi}{4})}$$

Step 3: Calculate the components

Now, let's calculate the x and y components of the vector: $$x = 10 \cos{(\frac{5\pi}{4})} \approx -7.071$$ $$y = 10 \sin{(\frac{5\pi}{4})} \approx -7.071$$

Step 4: Write the vector in component form

Finally, we can write the vector in component form using the calculated components: $$\mathbf{v} = <-7.071, -7.071>$$

Key Concepts

Converting Degrees to Radians

Converting the angle from degrees to radians is an essential step when working with vectors, as most mathematical functions in trigonometry use radians. Radians and degrees are both units for measuring angles, with one complete revolution being equal to 360 degrees, or \(2\pi\) radians. To convert from degrees to radians, we use the relationship that 180 degrees equals \(\pi\) radians. Thus, the formula for conversion is:
\[\text{radians} = \frac{\text{degrees} \times \pi}{180}\]
For example, to convert 225 degrees to radians, you multiply \(225\) by \(\frac{\pi}{180}\) to get \(\frac{5\pi}{4}\) radians. This conversion informs the trigonometric functions which comprise the vector's component form.

Polar-to-Cartesian Conversion

To describe the position of a vector accurately in a plane, we often convert its polar coordinates (magnitude and angle) to Cartesian coordinates (x, y). This is called polar-to-cartesian conversion. The conversion uses trigonometric functions, where the x-coordinate is found by multiplying the vector's magnitude by the cosine of its direction angle, and the y-coordinate by the sine of its direction angle. The formulas used are:
\[x = \| \mathbf{v} \| \cos{\theta}\]\[y = \| \mathbf{v} \| \sin{\theta}\]
This method transforms the vector's magnitude and direction into a precise location within the Cartesian coordinate system. This is particularly useful when combining vectors or projecting them onto axes.

Magnitude of a Vector

The magnitude of a vector, denoted as \(\|\mathbf{v}\|\), represents its length or size. When we speak about the 'magnitude', we are referring to the distance from the initial point of the vector to its terminal point when drawn on a graph or plane. It's calculated without considering its direction. When we have a vector in component form \(\mathbf{v} = \), its magnitude can be found using the Pythagorean theorem:
\[\|\mathbf{v}\| = \sqrt{x^2 + y^2}\]
When we start with the magnitude, like in the original problem, we use it to find the components of the vector in the Cartesian plane, as part of the polar-to-cartesian conversion process.

Direction Angle of a Vector

The direction angle of a vector, usually denoted as \(\theta\), is the angle the vector makes with the positive x-axis. The angle provides the direction of the vector and is an essential component when defining vectors in polar form or converting them to Cartesian coordinates. In the context of the original problem, the direction angle is given in degrees and needs to be converted to radians before applying trigonometric functions to find the Cartesian components. Once the vector is defined by its magnitude and direction angle, it can be uniquely positioned within a plane.
Knowing the direction angle also assists in graphically representing vectors and understanding their orientation relative to the coordinate axes.