Problem 41 - Algebra and Trigonometry

Question

\(41-46\) . Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors \(\mathbf{i}\) and \(\mathbf{j}\) . $$ |\mathbf{v}|=40, \quad \theta=30^{\circ} $$

Step-by-Step Solution

Answer

The vector is \( 20\sqrt{3}\mathbf{i} + 20\mathbf{j} \).

Step 1: Calculate Horizontal Component

The horizontal component of a vector can be found using the formula: \( v_x = | extbf{v}| imes ext{cos}(\theta) \). Here, \( | extbf{v}| = 40 \) and \( \theta = 30^{\circ} \). So, we calculate: \( v_x = 40 \times \text{cos}(30^{\circ}) = 40 \times \frac{\sqrt{3}}{2} = 20\sqrt{3} \).

Step 2: Calculate Vertical Component

The vertical component of a vector is found using the formula: \( v_y = | extbf{v}| imes ext{sin}(\theta) \). We substitute \( | extbf{v}| = 40 \) and \( \theta = 30^{\circ} \): \( v_y = 40 \times \text{sin}(30^{\circ}) = 40 \times \frac{1}{2} = 20 \).

Step 3: Express Vector in Terms of \( \mathbf{i} \) and \( \mathbf{j} \)

A vector in terms of \( \mathbf{i} \) and \( \mathbf{j} \) can be written as \( v_x \mathbf{i} + v_y \mathbf{j} \). Using the previously calculated components: \( v_x = 20\sqrt{3} \) and \( v_y = 20 \), the vector is \( 20\sqrt{3}\mathbf{i} + 20\mathbf{j} \).

Key Concepts

horizontal component

When dealing with vectors, finding the horizontal component is a crucial step. Imagine a vector as an arrow pointing in a specific direction, with a certain length (or magnitude). This direction has its horizontal and vertical influences. To find the horizontal component, we need to consider how far the arrow moves in the horizontal direction.

We use a simple trick from trigonometry. The formula for the horizontal component is given by \( v_x = |\mathbf{v}| \times \cos(\theta) \). Here, \( |\mathbf{v}| \) represents the length of the vector, and \( \theta \) is the angle it makes with the horizontal axis.

• If the vector has a length of 40 and an angle of 30 degrees, the calculation becomes \( v_x = 40 \times \cos(30^{\circ}) \).
• Using the trigonometric identity \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \), we find \( v_x = 40 \times \frac{\sqrt{3}}{2} = 20\sqrt{3} \).

This value tells us how much influence the vector has in the horizontal direction.

vertical component

The vertical component is another essential part of understanding vectors. This component tells us how much of the vector's influence is working in the vertical direction. Just like with the horizontal component, we use trigonometry to uncover its value.

We calculate the vertical component with the formula \( v_y = |\mathbf{v}| \times \sin(\theta) \). For our vector, where \( |\mathbf{v}| = 40 \) and \( \theta = 30^{\circ} \), the calculation is straightforward:

• Use \( \sin(30^{\circ}) = \frac{1}{2} \), then plug this into the formula: \( v_y = 40 \times \frac{1}{2} = 20 \).

This result shows the impact of the vector in the vertical direction. Together with the horizontal component, it helps us fully understand the vector's position and magnitude.

unit vector notation

Expressing vectors in unit vector notation is like giving directions with precise coordinates. This method uses the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), which represent the horizontal and vertical directions, respectively. Any vector can be broken down into a combination of these two vectors.

When we have the horizontal component \( v_x \) and the vertical component \( v_y \), the vector can be written as \( v_x \mathbf{i} + v_y \mathbf{j} \). For our vector example:

• The horizontal component was found to be \( 20\sqrt{3} \).
• The vertical component was \( 20 \).

This means our vector can be expressed as \( 20\sqrt{3} \mathbf{i} + 20 \mathbf{j} \).
This notation is a compact way to represent both magnitude and direction, making it easier to work with vectors in mathematical operations and various applications.