Problem 52 - Algebra and Trigonometry

Question

In Exercises \(47-52,\) write the vector \(\mathbf{v}\) in terms of i and \(\mathbf{j}\) whose magnitude lvi and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=\frac{1}{4}, \theta=200^{\circ}$$

Step-by-Step Solution

Answer

The vector \( \mathbf{v} \) is \( -\frac{1}{4} \cos(200^\circ) \mathbf{i} + \frac{1}{4} \sin(200^\circ) \mathbf{j} \). After simplification, this results in the final vector in terms of i and j.

Step 1: Convert angle into radians

Currently, the given angle \( \theta \) is in degrees. We will need to convert this into radians for our calculations. The conversion factor is \( \pi \) radians = 180 degrees. So, \( \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \).

Step 2: Compute the vector

Now that we have our angle in radians, we can input our values into our vector formula \( \mathbf{v} = \| \mathbf{v} \| \cos \theta \mathbf{i} + \| \mathbf{v} \| \sin \theta \mathbf{j} \) to get the vector \( \mathbf{v} \) in terms of i and j.

Step 3: Simplify the vector

After substituting the values, we will simplify the expressions to achieve the final vector components in terms of i and j.

Key Concepts

Magnitude

The magnitude of a vector is like the length of an arrow pointing in a certain direction. This value tells us how long the vector is without caring about the direction it points to. In mathematics, we represent the magnitude of a vector \( \mathbf{v} \) using the notation \( \| \mathbf{v} \| \).

For example, in the given exercise, the magnitude of the vector \( \mathbf{v} \) is \( \frac{1}{4} \). This means, if you were to imagine this vector on a graph, it would have a short "length." The magnitude is always a non-negative value because length cannot be negative.

Some key points to remember about magnitude:

• The magnitude indicates the size or extent of the vector.
• Magnitude can be found using the Pythagorean theorem for vectors with known components.
• It is a scalar quantity, which means it has magnitude only, not direction.

Direction Angle

The direction angle of a vector specifies which way a vector is pointing. It's like pointing an arrow in a specific direction on a map. This angle is measured from the positive x-axis (usually in a counterclockwise direction).

In terms of our exercise, we have a direction angle \( \theta = 200^{\circ} \). This tells us that the vector is pointing 200 degrees away from the positive x-axis. It’s important to convert degrees to radians when performing trigonometric calculations because mathematical functions in calculus often require angular input in radians.

It's helpful to remember that:

• The direction angle defines the orientation of the vector.
• To convert degrees to radians, multiply the degree value by \( \frac{\pi}{180} \), because \( 180^{\circ} = \pi \) radians.
• A direction angle can range from \( 0^{\circ} \) to \( 360^{\circ} \) in circle geometry.

Trigonometric Functions

Trigonometric functions are used to relate the angles and sides of a triangle, but they are also crucial in vector representation. When a direction angle is given, trigonometry helps in breaking a vector down into its components along the x and y axes.

For the vector \( \mathbf{v} \), we use:- \( \cos \theta \) to find the component along the x-axis- \( \sin \theta \) to find the component along the y-axis

So in the exercise, for a vector \( \mathbf{v} \) with angle \( 200^{\circ} \), calculate these using your converted angle in radians. If \( \theta \) was changed to radians, use \( \theta_{\text{rad}} \) in your formula:

• \( v_x = \| \mathbf{v} \| \cos \theta_{\text{rad}} \)
• \( v_y = \| \mathbf{v} \| \sin \theta_{\text{rad}} \)

Trigonometric functions are fundamental tools in finding these components, which are essential for forming the vector in terms of \( \mathbf{i} \) and \( \mathbf{j} \).

Unit Vectors

Unit vectors provide a way to express a vector's direction and are key in breaking vectors into components. A unit vector has a magnitude of 1 and indicates direction only, not length.

In Cartesian coordinates:- The unit vector \( \mathbf{i} \) represents the x-direction, with components \( (1, 0) \).- The unit vector \( \mathbf{j} \) represents the y-direction, with components \( (0, 1) \).

Together, these unit vectors offer a way to specify any vector as a combination of these two directions. For example, if a vector is written as:\( \mathbf{v} = a \mathbf{i} + b \mathbf{j} \)This vector has components \( a \) in the x-direction and \( b \) in the y-direction.

• Unit vectors reduce complex vector expressions to simplified format.
• They are dimensionless, serving solely for directional purposes.
• Unit vectors in vector decomposition help make understanding vector components intuitive.

Through these unit vectors, we can describe the direction and magnitude of any vector in a straightforward manner.