Problem 31 - Algebra and Trigonometry

Question

Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and j. $$ |\mathbf{v}|=4, \quad \theta=10^{\circ} $$

Step-by-Step Solution

Answer

\( \mathbf{v} \approx 3.9392 \mathbf{i} + 0.6944 \mathbf{j} \)

Step 1: Identify the Components

The horizontal and vertical components of a vector can be found using trigonometric functions. For a vector \( \mathbf{v} \) with magnitude \( |\mathbf{v}| \) and direction \( \theta \):- The horizontal component is found using \( |\mathbf{v}| \cos(\theta) \).- The vertical component is found using \( |\mathbf{v}| \sin(\theta) \).

Step 2: Calculate the Horizontal Component

The horizontal component is given by \[x = |\mathbf{v}| \cos(\theta) = 4 \cos(10^{\circ})\]Using a calculator, \( \cos(10^{\circ}) \approx 0.9848 \).Thus, the horizontal component is:\[x = 4 \times 0.9848 \approx 3.9392\]

Step 3: Calculate the Vertical Component

The vertical component is given by \[y = |\mathbf{v}| \sin(\theta) = 4 \sin(10^{\circ})\]Using a calculator, \( \sin(10^{\circ}) \approx 0.1736 \).Thus, the vertical component is:\[y = 4 \times 0.1736 \approx 0.6944\]

Step 4: Express the Vector Using Basis Vectors

Now that we have the components, we can write the vector \( \mathbf{v} \) in terms of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) as follows:\[\mathbf{v} = 3.9392 \mathbf{i} + 0.6944 \mathbf{j}\]

Key Concepts

Trigonometric Functions

Understanding trigonometric functions is key to solving problems related to vector components. Trigonometric functions include sine, cosine, and tangent. These functions help us relate angles to sides in right-angled triangles. When you have a direction angle \( \theta \) of a vector, these functions can determine how the vector spreads along the horizontal and vertical axes. For example:

• Cosine (\( \cos \)) relates the angle's adjacent side (horizontal component) to the hypotenuse (the vector's magnitude).
• Sine (\( \sin \)) concerns the angle's opposite side (vertical component) and the hypotenuse.

By using these functions, it's possible to decompose any vector into its horizontal and vertical parts. Essentially, \[x = |\mathbf{v}| \cos(\theta)\] for horizontal and \[y = |\mathbf{v}| \sin(\theta)\] for vertical.

Horizontal Component

The horizontal component of a vector points along the x-axis. This component reflects how far a vector moves sideways, usually quantified from a vector’s starting point. By using trigonometric functions like cosine, we can easily calculate it. If you have a vector with magnitude \( |\mathbf{v}| \) and angle \( \theta \), the expression \[x = |\mathbf{v}| \cos(\theta)\] will give the horizontal component.To find this value, input the angle into the cosine function using a calculator. In our example, with \( |\mathbf{v}| = 4 \) and \( \theta = 10^{\circ} \), it's calculated as: - \( x = 4 \times 0.9848 \approx 3.9392 \), where \( \cos(10^{\circ}) \approx 0.9848\). This value tells us how much the vector stretches along the horizontal.

Vertical Component

The vertical component signifies how much a vector extends upward or downward along the y-axis. To isolate this part of a vector, trigonometric sine is employed, linking the vector's magnitude and angle: \[y = |\mathbf{v}| \sin(\theta)\].For our particular case, we have a magnitude of 4 and an angle of \(10^{\circ}\). To find the vertical component, you:- Use the sine function: \( \sin(10^{\circ}) \approx 0.1736 \).- Calculate: \( y = 4 \times 0.1736 \approx 0.6944 \).This tells us the vector’s vertical movement from its origin point. The result is clear: an approximation of how much the vector climbs.

Unit Vectors i and j

In vector notation, unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) are crucial for expressing vectors in two-dimensional space. Unit vectors are vectors with a magnitude of one, essentially serving as foundational direction pointers along axes:

• \( \mathbf{i} \) points along the positive x-axis.
• \( \mathbf{j} \) aligns with the positive y-axis.

These unit vectors allow us to express any vector in terms of its horizontal and vertical components. For example, if you have calculated \( x \approx 3.9392 \) and \( y \approx 0.6944 \) for the vector \( \mathbf{v} \), it can be rewritten using unit vectors as:\[ \mathbf{v} = 3.9392 \mathbf{i} + 0.6944 \mathbf{j} \]. This straightforward expression clearly details both small parts of the vector, preserving direction and magnitude when it comes to spatial representation.