Problem 77 - Precalculus with Limits

Question

Einding the Component Form of a Vector In Exercises \(75-78\) , find the component form of the sum of u and v with direction angles \(\theta_{\text { u }}\) and \(\theta_{v}\) . $$\begin{array}{ll}{\text { Magnitude }} & {\text { Angle }} \\\ {\|\mathbf{u}\|=20} & {\theta_{\mathrm{u}}=45^{\circ}} \\ {\|\mathbf{v}\|=50} & {\theta_{\mathrm{v}}=180^{\circ}}\end{array}$$

Step-by-Step Solution

Answer

The component form of the sum of vectors u and v is \((x_{sum}, y_{sum})\).

Step 1: Find Components of Vector u

Using the magnitude and direction angle of vector u, compute its x and y components. Use these formulas:\n\n\(x_{u} = \|u\| \cos(\theta_u)= 20 \cos(45^{\circ})\)\n\(y_{u} = \|u\| \sin(\theta_u)= 20 \sin(45^{\circ})\)

Step 2: Find Components of Vector v

Similarly, find the x and y components of vector v:\n\n\(x_{v} = \|v\| \cos(\theta_v)= 50 \cos(180^{\circ})\)\n\(y_{v} = \|v\| \sin(\theta_v)= 50 \sin(180^{\circ})\)

Step 3: Compute the Sum of the Components

Add the respective components of the two vectors:\n\n\(x_{sum} = x_u + x_v\)\n\(y_{sum} = y_u + y_v\)\nThese will give the x and y components of the resultant vector.

Key Concepts

Component Form

Vectors can be expressed in component form, which is essentially breaking down a vector into its horizontal and vertical components along the x and y axes. Understanding component form is crucial when dealing with vector summation, as it allows for a straightforward addition of vectors.

• The component form of a vector \( \mathbf{u} \) is given by the coordinates \((x_u, y_u)\), where \( x_u \) and \( y_u \) are obtained using trigonometric functions.
• To find these components, you use the magnitude \( \|\mathbf{u}\| \) and the direction angle \( \theta_u \):
  • \( x_u = \|\mathbf{u}\| \cos(\theta_u) \)
  • \( y_u = \|\mathbf{u}\| \sin(\theta_u) \)

In practical terms, you just consider how the vector projects onto the x and y axes. By calculating these projections, you have effectively converted the vector into a form that is easy to work with for addition or further analysis.

Direction Angles

Direction angles give you the orientation of a vector in a plane. In two dimensions, the direction angle \( \theta \) is the angle between the positive x-axis and the vector. Knowing this angle helps in decomposing the vector into components.

• Direction angle \( \theta \) is used in conjunction with trigonometric functions (cosine and sine) to find a vector's components.
• A direction angle can range from \(0^{\circ}\) to \(360^{\circ}\) and indicates the counter-clockwise rotation from the positive x-axis.
• Using this angle, you can calculate:
  • \( x = \text{magnitude} \times \cos(\theta) \)
  • \( y = \text{magnitude} \times \sin(\theta) \)

Understanding direction angles is vital as they determine how far and in what direction a vector points. In addition problems, you might need to find these components before summing vectors together to easily determine the resultant vector's direction.

Vector Magnitude

The magnitude of a vector is simply its length. It's a measure of how "strong" or how "far" the vector reaches in its directional path. This is a crucial part of vector representation since the magnitude works in tandem with direction to define a vector completely.

• To find the length of a vector, use the Pythagorean theorem, considering its components: \( \|\mathbf{v}\| = \sqrt{x^2 + y^2} \).
• The magnitude is always a non-negative number and provides necessary information for operations like normalization and scalar multiplication.
• In the context of addition or arithmetic operations, the magnitude helps adjust the influence of each vector's contributions to the resultant vector.

When using vectors in physics, engineering, or graphics, understanding their magnitude becomes crucial as it directly impacts forces, displacements, and velocities. You compute it using functions that break down a vector into its parts, reflecting how much "force" it represents in both the x and y directions.