Problem 30 - Contemporary Precalculus

Question

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=20, \theta=120^{\circ}$$

Step-by-Step Solution

Answer

Answer: The component form of the vector is $$\langle -10, 10\sqrt{3} \rangle$$.

Step 1: (Step 1: Find the horizontal component (x-component) of the vector.)

To find the horizontal component, we can use the formula: $$v_x = \|v\| cos(\theta)$$ We know that \(\|v\|=20\) and \(\theta=120^{\circ}\). We need to find \(cos(120^{\circ})\). Step 2: Calculate \(cos(120^{\circ})\).

Step 2: (Step 2: Calculate \(cos(120^{\circ})\).)

We can find the value of \(cos(120^{\circ})\) from a calculator or a reference table. $$cos(120^{\circ}) = -\frac{1}{2}$$ Step 3: Calculate the horizontal component (x-component) of the vector.

Step 3: (Step 3: Calculate the horizontal component (x-component) of the vector.)

Now, we have everything to calculate the x-component of the vector. $$v_x = \|v\| cos(\theta) = 20(-\frac{1}{2}) = -10$$ Step 4: Find the vertical component (y-component) of the vector.

Step 4: (Step 4: Find the vertical component (y-component) of the vector.)

To find the vertical component, we can use the formula: $$v_y = \|v\| sin(\theta)$$ We know that \(\|v\|=20\) and \(\theta=120^{\circ}\). We need to find \(sin(120^{\circ})\). Step 5: Calculate \(sin(120^{\circ})\).

Step 5: (Step 5: Calculate \(sin(120^{\circ})\).)

We can find the value of \(sin(120^{\circ})\) from a calculator or a reference table. $$sin(120^{\circ}) = \frac{\sqrt{3}}{2}$$ Step 6: Calculate the vertical component (y-component) of the vector.

Step 6: (Step 6: Calculate the vertical component (y-component) of the vector.)

Now, we have everything to calculate the y-component of the vector. $$v_y = \|v\| sin(\theta) = 20(\frac{\sqrt{3}}{2}) = 10\sqrt{3}$$ Step 7: Write the component form of the vector \(v\).

Step 7: (Step 7: Write the component form of the vector \(v\).)

We have found the horizontal and vertical components of the vector \(v\). The component form is: $$\mathbf{v} = \langle v_x, v_y \rangle = \langle -10, 10\sqrt{3} \rangle$$

Key Concepts

Magnitude of a Vector

The 'magnitude of a vector' is like the vector's length. It tells us how long the vector is, and is often denoted as \( \|v\| \). For example, when a vector has a magnitude of 20, it means the vector stretches out to a length of 20 units. To visualize this, imagine the vector as an arrow on a graph. The length of this arrow shows how far it reaches. Knowing the magnitude helps us calculate the other parts of the vector, such as the 'trigonometric components' (which are the x and y components). We use the magnitude along with the direction angle to get a more complete picture of the vector's position and direction on the graph.

Direction Angle

The 'direction angle' of a vector specifies the direction in which the vector is pointing. For instance, if a vector has a direction angle of \(120^\circ\), this angle is measured counterclockwise from the positive x-axis. Visualize this angle on a unit circle where \(0^\circ\) is right along the positive x-axis. As the angle increases, the direction changes in a counter-clockwise manner:

• \(0^\circ\) points to the right, along the x-axis.
• \(90^\circ\) points straight up, along the y-axis.
• \(180^\circ\) points to the left, opposite \(0^\circ\).

Thus, a \(120^\circ\) direction indicates a vector that is slightly left of straight up. The direction angle, combined with the vector’s magnitude, helps find the vector’s trigonometric components.

Trigonometric Components

'Trigonometric components' refer to breaking down a vector into its horizontal and vertical parts. These components are essential in representing how much of the vector extends in the x and y directions.To find these components, we use trigonometry:

• The x-component uses cosine: \( v_x = \|v\| \cos(\theta) \).
• The y-component uses sine: \( v_y = \|v\| \sin(\theta) \).

These formulas originate from right triangle trigonometry, where a vector can be considered the hypotenuse of a triangle. Understanding these components is crucial, as they allow the precise placement of the vector on a coordinate grid. Furthermore, these components provide the vector's exact impact in each direction.

X-component

The 'x-component' (\( v_x \)) of a vector shows how much the vector goes along the x-axis or horizontally. To calculate it, we multiply the vector's magnitude by the cosine of its direction angle: \( v_x = \|v\| \cos(\theta) \).In our example, with a magnitude of 20 and a direction angle of \(120^\circ\), we found that \( \cos(120^\circ) = -\frac{1}{2} \). Thus, the x-component becomes:\[ v_x = 20 \times \left(-\frac{1}{2}\right) = -10 \]This negative value signifies that the vector is pointing to the left, reflecting the vector’s overall direction. Understanding the x-component allows us to grasp how far and in which direction the vector travels horizontally.

Y-component

The 'y-component' (\( v_y \)) of a vector indicates how much the vector stretches in the vertical or y-axis direction. It is calculated as the product of the vector's magnitude and the sine of its direction angle: \( v_y = \|v\| \sin(\theta) \).For our specific case, given a magnitude of 20 and a direction angle of \(120^\circ\), we determined that \( \sin(120^\circ) = \frac{\sqrt{3}}{2} \). This leads us to calculate the y-component as:\[ v_y = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3} \]This positive value shows that the vector extends upward along the y-axis. Understanding the y-component is vital, as it tells us how much influence the vector has in the vertical direction.