Problem 29 - Precalculus

Question

Find the vector, given its magnitude and direction angle. $$|\mathbf{u}|=4, \theta=310^{\circ}$$

Step-by-Step Solution

Answer

The vector is \(\langle 2.571, -3.064 \rangle\).

Step 1: Identify the Problem

We are given a vector's magnitude and a direction angle, and we need to find the vector in component form.

Step 2: Understand the Relationship

A vector can be represented as \(uzzy\) where \(v_x\) is the horizontal component and \(v_y\) is the vertical component. These can be found using the magnitude \(|47|\) and direction angle \(7enne}, \(v_x = D(\0er} \) and 310_{x}\).

Step 3: Calculate Horizontal Component

The horizontal component \(v_x\) of the vector can be calculated as \(|\mathbf{u}| \cdot \cos\theta\). Substituting the values, we have: \[v_x = 4 \cdot \cos(310^{\circ})= 4 \cdot \cos(-50^{\circ})\] \[v_x \approx 4 \cdot 0.6428 \approx 2.571\]

Step 4: Calculate Vertical Component

The vertical component \(v_y\) of the vector can be calculated as \(|\mathbf{u}| \cdot \sin\theta\). Substituting the values, we have: \[v_y = 4 \cdot \sin(310^{\circ}) = 4 \cdot \sin(-50^{\circ})\]\[v_y \approx 4 \cdot (-0.7660) \approx -3.064\]

Step 5: Combine Components

Now that we have both components, the vector \(\mathbf{u}\) in component form is \(\langle v_x, v_y \rangle\). Therefore, the vector is:\[\mathbf{u} = \langle 2.571, -3.064 \rangle\]

Key Concepts

Magnitude and Direction

Vectors have two main characteristics: magnitude and direction. The magnitude is essentially the "length" or "size" of the vector, telling you how far the vector extends. In simpler terms, if you think of a vector as an arrow, the magnitude would be how long the arrow is.

Direction, on the other hand, specifies where the vector is pointing. Typically, the angle of direction is measured from a standard position (often the positive x-axis) in a counter-clockwise direction. In the exercise provided, the vector direction is given as 310 degrees. This is somewhat unique because it's measured clockwise from the positive x-axis, an approach sometimes used in physics and engineering.

• Magnitude in this case: 4 (units of your choice)
• Direction: 310 degrees, usually measured clockwise for certain conventions

To understand this fully, picture starting at the positive x-axis and moving clockwise until you reach the angle of 310 degrees. Think of it as starting from a flat horizontal line and slanting it downward slightly as you reach that angle.

Component Form

The component form of a vector breaks it down into its basic horizontal (x) and vertical (y) components. Think of it like separating the vector's effects into two directions – how much it moves horizontally and how much it moves vertically.

Using the magnitude and direction, these components can be calculated using trigonometric functions:

• The horizontal component (\(v_x\)): This tells you how much the vector moves along the x-axis.
• The vertical component (\(v_y\)): This tells you how much the vector moves along the y-axis.

Once you compute these, you can represent your vector as \(\langle v_x, v_y \rangle\). This makes it easy to understand the vector's trajectory and effects in practical applications, like physics or engineering problems. In this exercise, the vector's component form is \(\langle 2.571, -3.064 \rangle\), derived using the cosine and sine of the direction angle with respect to its magnitude.

Trigonometric Functions

Trigonometric functions play a crucial role in vector representation, especially when converting from magnitude and direction to component form. Here’s how they work:

• Cosine function (\(\cos\theta\)): This gives the ratio of the adjacent side to the hypotenuse in a right-angled triangle. For vectors, it helps find the horizontal component.
• Sine function (\(\sin\theta\)): This gives the ratio of the opposite side to the hypotenuse. In vectors, it assists in calculating the vertical component.

For example, given the direction angle \(\theta = 310^\circ\), the horizontal component \(v_x\) is \(|\mathbf{u}| \cdot \cos(310^\circ)\) and the vertical component \(v_y\) becomes \(|\mathbf{u}| \cdot \sin(310^\circ)\).

It’s important to convert the angle to its equivalent in the standard trigonometric framework if dealing with angles beyond \(360^\circ\) or negative angles, ensuring accurate computation. By understanding these functions, you can adeptly switch between different forms of vector representation, enhancing your analytical tools in mathematics and related fields.