Problem 78 - 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 \(\theta_{v}\) . $$\begin{array}{ll}{\text { Magnitude }} & {\text { Angle }} \\\ {\|\mathbf{u}\|=50} & {\theta_{\mathrm{u}}=30^{\circ}} \\ {\|\mathbf{v}\|=30} & {\theta_{x}=110^{\circ}}\end{array}$$

Step-by-Step Solution

Answer

The component form of the sum of u and v with the given direction angles is \((u+v)_{x}, (u+v)_{y}\).

Step 1: Convert Polar Coordinates to Cartesian Coordinates

The conversion of polar coordinates (magnitude, direction) to Cartesian coordinates (x, y) is performed using the formulas: \[x = r \cdot cos(\theta)\] , \[y = r \cdot sin(\theta)\] . Apply these formula for vector u and vector v to get Cartesian coordinates.

Step 2: Find Coordinate Form of Vectors u and v

We can represent a vector in 2D space using two coordinates (x,y). These correspond to the x-component (horizontal) and y-component (vertical) of the vector respectively. For vector u, the coordinates would be (u_x, u_y) and for vector v, the coordinates would be (v_x, v_y). Calculate these values using the conversion formulas from step 1.

Step 3: Find Vector Sum

The sum of two vectors is performed by adding the corresponding components of the vectors. This means that the x-component of the sum of vector u and v (\(u+v\)) is equal to \(u_x + v_x\) and the y-component is equal to \(u_y + v_y\).\nCalculate these values to find the vector sum.

Step 4: State the Result in Component Form

The vector sum \(u+v\) can now be given in component form as \((u+v)_{x}, (u+v)_{y}\). This is the final answer.

Key Concepts

Component Form of Vectors

When dealing with vectors, the component form is one of the most useful ways to express them. Vectors have both magnitude and direction, and in the component form, they are broken down into their horizontal (x) and vertical (y) components.

• X-component: Represents the horizontal direction.
• Y-component: Represents the vertical direction.

These components can be added or subtracted to perform various vector operations. For instance, if you have vectors \(\mathbf{u}\) and \(\mathbf{v}\) with components \((u_x, u_y)\) and \((v_x, v_y)\) respectively, their sum is \((u_x+v_x, u_y+v_y)\). This makes it easier to visualize and compute vector problems.

Direction Angles of Vectors

Direction angles describe the orientation of a vector in the plane. It is the angle that a vector makes with the positive x-axis. This concept is important for converting vectors between forms.

• Measure in Degrees: Typically measured in degrees.
• Conversion to Radians: May sometimes need conversion to radians for trigonometric calculations, using \(\theta_{\text{radians}} = \theta_{\text{degrees}} \times \left(\frac{\pi}{180}\right)\).

Calculating the direction angle gives insight into the vector's path and helps in finding the vector's components through trigonometric functions like cosine and sine.

Understanding Polar Coordinates

Polar coordinates represent a point in terms of its distance from the origin and the angle from the positive x-axis.

The polar form of a vector is given by its magnitude (\(r\)) and angle (\(\theta\)):

• Magnitude \((r):\) The distance from the origin to the point.
• Angle \((\theta):\) The angle formed with the positive x-axis.

To transform polar coordinates into cartesian coordinates, use the formulas:

• \(x = r \cdot \cos(\theta)\)
• \(y = r \cdot \sin(\theta)\)

These transformations are essential for visualizing and computing vector operations in a standard coordinate plane.

Converting to Cartesian Coordinates

Cartesian coordinates describe a point in terms of x and y on a two-dimensional plane. This form is straightforward for vector operations like addition, subtraction, and multiplication.

• X-coordinate \((x):\) Uses the cosine of the angle, representing horizontal distance.
• Y-coordinate \((y):\) Uses the sine of the angle, representing vertical distance.

By converting vectors from polar to Cartesian form, calculations such as finding the component form become manageable. For example, knowing a vector's magnitude and direction angle allows one to determine the exact position of a vector in terms of x and y, facilitating further mathematical operations like finding a vector's sum in component form.