Problem 32 - 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. $$ |\mathbf{v}|=\sqrt{3}, \quad \theta=300^{\circ} $$

Step-by-Step Solution

Answer

The vector is \(\mathbf{v} = \frac{\sqrt{3}}{2} \mathbf{i} - \frac{3}{2} \mathbf{j}\).

Step 1: Understanding the Components

To break the vector into its horizontal and vertical components, we need to use trigonometric functions: cosine for the horizontal component and sine for the vertical component. Given the vector length \(|\mathbf{v}| = \sqrt{3}\) and the angle \(\theta = 300^{\circ}\).

Step 2: Calculate the Horizontal Component

The horizontal component \(v_x\) of the vector is calculated using the formula \(v_x = |\mathbf{v}| \cdot \cos(\theta)\). Substituting the given values, we have \(v_x = \sqrt{3} \cdot \cos(300^{\circ})\). Since \(\cos(300^{\circ}) = \frac{1}{2}\), we find \(v_x = \sqrt{3} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2}\).

Step 3: Calculate the Vertical Component

The vertical component \(v_y\) of the vector is calculated using the formula \(v_y = |\mathbf{v}| \cdot \sin(\theta)\). Substituting the given values, we have \(v_y = \sqrt{3} \cdot \sin(300^{\circ})\). Since \(\sin(300^{\circ}) = -\frac{\sqrt{3}}{2}\), we find \(v_y = \sqrt{3} \cdot -\frac{\sqrt{3}}{2} = -\frac{3}{2}\).

Step 4: Writing the Vector in Terms of \(\mathbf{i}\) and \(\mathbf{j}\)

The vector \(\mathbf{v}\) can be written in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) (representing the horizontal and vertical unit vectors, respectively) using the components we found: \(\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j}\). So, using \(v_x = \frac{\sqrt{3}}{2}\) and \(v_y = -\frac{3}{2}\), we have \(\mathbf{v} = \frac{\sqrt{3}}{2} \mathbf{i} - \frac{3}{2} \mathbf{j}\).

Key Concepts

Trigonometric Functions

When dealing with vectors, trigonometric functions like cosine and sine are extremely helpful. They help us decompose a vector into its horizontal and vertical components.
\[ \text{Cosine} \theta \] is used to determine the horizontal component of a vector. This tells us how much the vector extends along the x-axis.
Similarly, \[ \text{Sine} \theta \] gives us the vertical component, indicating how much the vector extends along the y-axis.
These functions are based on right triangles. The cosine of an angle is the ratio of the adjacent side to the hypotenuse, while sine is the ratio of the opposite side to the hypotenuse.

• Cosine: \( v_x = |\mathbf{v}| \cdot \cos(\theta) \)
• Sine: \( v_y = |\mathbf{v}| \cdot \sin(\theta) \)

Understanding how these functions relate to vector components is crucial for accurately breaking down vectors.

Unit Vectors

Unit vectors are handy tools in vector mathematics. They have a magnitude of one and are used to indicate direction in a coordinate system.
Two common unit vectors in a 2D Cartesian plane are \( \mathbf{i} \) and \( \mathbf{j} \).

• \( \mathbf{i} \) represents the x-axis direction (horizontal).
• \( \mathbf{j} \) represents the y-axis direction (vertical).

When expressing a vector using its components, we often write it in terms of these unit vectors. For instance, a vector \( \mathbf{v} \) with components \( v_x \) and \( v_y \) can be expressed as \( v_x \cdot \mathbf{i} + v_y \cdot \mathbf{j} \). This notation makes it easy to visualize and calculate vector direction and length.

Horizontal Component

The horizontal component of a vector tells us how much it stretches along the x-axis. This is crucial when determining the vector's full direction and impact.
To find this component, we use the formula \( v_x = |\mathbf{v}| \cdot \cos(\theta) \).
Using cosine helps us measure the adjacent side of the angle in a right triangle. For our given vector:

• \( |\mathbf{v}| = \sqrt{3} \) and \( \theta = 300^{\circ} \)
• The calculation is \( v_x = \sqrt{3} \cdot \cos(300^{\circ}) \)
• \( \cos(300^{\circ}) = \frac{1}{2} \), so \( v_x = \frac{\sqrt{3}}{2} \)

This horizontal component indicates the proportion of the vector's length acting in the x-direction.

Vertical Component

The vertical component shows how much of the vector extends along the y-axis. This component is equally important to determine the vector's direction and effect.
It can be found using the formula \( v_y = |\mathbf{v}| \cdot \sin(\theta) \).
Since sine relates to the opposite side in a right triangle, it helps us capture vertical movement. For our vector:

• Given \( |\mathbf{v}| = \sqrt{3} \) and \( \theta = 300^{\circ} \)
• The calculation is \( v_y = \sqrt{3} \cdot \sin(300^{\circ}) \)
• \( \sin(300^{\circ}) = -\frac{\sqrt{3}}{2} \), hence \( v_y = -\frac{3}{2} \)

This vertical component indicates how the vector influences the y-axis negatively, implying a downward direction.