Problem 75 - Precalculus with Limits

Question

In Exercises 75-78, find the component form of the sum of \(\mathbf{u}\) and \(\mathbf{v}\) with direction angles \(\mathbf{\theta_u}\) and \(\mathbf{\theta_v}\). Magnitude ||\(\small{\mathbf{u}}\)|| \(= 5\) ||\(\small{\mathbf{v}}\)|| \(= 5\) Angle \(\mathbf{\theta_u} = 0^{\circ}\) \(\mathbf{\theta_v} = 90^{\circ}\)

Step-by-Step Solution

Answer

The component form of the sum of \(\mathbf{u}\) and \(\mathbf{v}\) is (5, 5).

Step 1: Convert from polar to Cartesian coordinates

First, convert both vectors from polar coordinates to Cartesian coordinates. For vector \(\mathbf{u}\), the x-component can be found by multiplying the magnitude of \(\mathbf{u}\) by \(cos(\theta_u)\), and the y-component can be found by multiplying the magnitude of \(\mathbf{u}\) by \(sin(\theta_u)\). The same process can be repeated for vector \(\mathbf{v}\).

Step 2: Compute Cartesian coordinates

Compute the Cartesian coordinates. For \(\mathbf{u}\), it comes out to \((5 cos 0^{\circ}, 5 sin 0^{\circ}) = (5, 0)\) and for \(\mathbf{v}\) it is \((5 cos 90^{\circ}, 5 sin 90^{\circ}) = (0, 5)\).

Step 3: Add the vectors

Add the two vectors together in Cartesian form. The sum of the x-components is the x-component of the result, and the sum of the y-components is the y-component of the result. So, the sum of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is \((5 + 0, 0 + 5) = (5, 5)\).

Key Concepts

Polar to Cartesian Coordinates

When dealing with vectors, one common method is to convert from polar coordinates to Cartesian coordinates. This makes it easier to perform vector operations like addition. Polar coordinates describe a vector in terms of its magnitude and direction angle. To convert to Cartesian coordinates, which describe the vector in terms of x and y components:

• Calculate the x-component as the magnitude of the vector times the cosine of the direction angle: \(x = r \cdot \cos(\theta)\)
• Calculate the y-component as the magnitude of the vector times the sine of the direction angle: \(y = r \cdot \sin(\theta)\)

In our exercise, for a vector \(\mathbf{u}\) with magnitude 5 and angle \(0^{\circ}\), the Cartesian coordinates are \((5,0)\). For \(\mathbf{v}\) with magnitude 5 and angle \(90^{\circ}\), they are \((0,5)\). This conversion is crucial for adding vectors efficiently.

Magnitude and Direction Angles

Understanding magnitude and direction angles is essential for working with vectors. The magnitude represents the length or size of the vector. Meanwhile, the direction angle shows where the vector is pointing.

• The angle is measured from the positive x-axis.
• If the angle is \(0^{\circ}\), the vector points directly along the x-axis.
• If it's \(90^{\circ}\), it points directly up along the y-axis.

In this exercise, \(\mathbf{u}\) has a magnitude of 5 and a direction angle of \(0^{\circ}\). It lies flat along the x-axis. Vector \(\mathbf{v}\) also has a magnitude of 5 but points straight up with a direction angle of \(90^{\circ}\). Recognizing these angles helps in visualizing and combining vectors.

Component Form

Vectors in component form are broken down into their x and y parts, giving a clear view of their direction and magnitude in a coordinate system. This form is useful because it simplifies vector operations.

• Each vector is expressed as \((x, y)\), where \(x\) and \(y\) are the components.
• Operations like addition and subtraction can be performed component-wise.

For example, in our exercise, vector \(\mathbf{u}\) is \((5,0)\) and vector \(\mathbf{v}\) is \((0,5)\). Their component form directly tells you that \(\mathbf{u}\) moves 5 units along the x-axis, while \(\mathbf{v}\) moves 5 units along the y-axis. Understanding component form is pivotal for later transformations and calculations.

Cartesian Form of Vectors

The Cartesian form of a vector is an algebraic representation using coordinates. This makes vectors easy to visualize in terms of their x and y movements in a plane.

• In Cartesian form, vectors are expressed as \(\mathbf{v} = (x,y)\).
• Addition of vectors in Cartesian form involves simple addition of their respective components, i.e., add x-components together and y-components together.

In our exercise, the vectors \(\mathbf{u} = (5,0)\) and \(\mathbf{v} = (0,5)\) are summed to \((5+0, 0+5) = (5,5)\). The resulting vector in Cartesian form, \((5,5)\), indicates a movement of 5 units along the x-axis and 5 units along the y-axis, providing a straightforward multidimensional representation.