Problem 49 - Algebra and Trigonometry

Question

In Exercises \(47-52,\) write the vector \(\mathbf{v}\) in terms of i and \(\mathbf{j}\) whose magnitude livì and direction angle \(\theta\) are given. $$|\mathbf{v}|=12, \theta=225^{\circ}$$

Step-by-Step Solution

Answer

The vector in terms of i and j, with magnitude 12 and direction \(225^{\circ}\) is \(\mathbf{v} = 12 \cos (225^{\circ})\, \mathbf{i} + 12 \sin (225^{\circ})\,\mathbf{j}\)

Step 1: Compute the i component of the vector

The i component of the vector \(\mathbf{v}\) can be calculated by multiplying the magnitude of the vector by the cosine of its direction angle. Using this formula results in \(v_{i} = |\mathbf{v}| \cos (\theta) = 12 \cos (225^{\circ})\)

Step 2: Compute the j component of the vector

Now, to find the j component, it is necessary to multiply the magnitude of the vector by the sine of its direction angle: \(v_{j} = |\mathbf{v}| \sin (\theta) = 12 \sin (225^{\circ})\)

Step 3: Write down the vector \(\mathbf{v}\)

Now that we have the i and j components of \(\mathbf{v}\), we can write down \(\mathbf{v}\) in terms of i and \(\mathbf{j}\): \(\mathbf{v} = v_{i} \mathbf{i} + v_{j} \mathbf{j}\)

Key Concepts

Magnitude of a Vector

In the realm of vectors, the **magnitude** represents the length or size of the vector. You can think of it as the distance from the origin to the point represented by the vector in a coordinate system. In mathematical terms, for a vector \(mathbf{v}\), the magnitude is denoted as \(|mathbf{v}|\).
To find the **magnitude**, you use the formula
\[|mathbf{v}| = \sqrt{{v_i}^2 + {v_j}^2}\] for a 2D vector with components \(v_i\) and \(v_j\).
The magnitude gives us insights into how large or small the vector is, without concern for its direction.

• If you have a vector's magnitude along with its direction, you can reconstruct the vector's components.
• For instance, if a vector has a magnitude of \(12\) units, regardless of direction, its length is \(12\) units.

Understanding magnitude is crucial when talking about vector lengths, force strengths, or velocity speeds.

Direction Angle

The **direction angle** of a vector tells us which direction the vector is pointing. It's measured from the positive x-axis (rightward direction on a graph) and is usually given in degrees or radians.
This angle helps define the direction of our vector in the plane. The full circle is 360 degrees, so an angle of 225 degrees points somewhere in the third quadrant.
Here's how to visualize it:

• If \(\theta = 0^{\circ}\), the vector points directly along the positive x-axis.
• If \(\theta = 90^{\circ}\), it points upwards along the positive y-axis.
• With \(\theta = 225^{\circ}\), the vector points diagonally into the lower left part of the graph, crossing both negative x and negative y axes.

Knowing the direction angle helps in understanding the orientation of the path or force represented by the vector.

Vector Components

**Vector components** break down the vector into parts along the axes of the coordinate plane, usually the x-axis and y-axis in two dimensions. When a vector has a given magnitude and direction, we use trigonometric functions to determine its components:
- The **i component** aligns with the x-axis and is computed as \(v_i = |mathbf{v}| \cos(\theta)\). - The **j component** aligns with the y-axis and is found by \(v_j = |mathbf{v}| \sin(\theta)\).
These components are important because they tell us exactly how much a vector moves in x and y directions.
Thereby, you can represent a 2D vector \(\mathbf{v}\) as
\(\mathbf{v} = v_i \mathbf{i} + v_j \mathbf{j}\) where \(\mathbf{i}\) and \(\mathbf{j}\) are the unit vectors along the x and y axes respectively.

This breakdown is handy for analysis as it simplifies calculations and helps in understanding complex movements or forces in their simplest forms.