Problem 51 - Precalculus with Limits

Question

In Exercises 49-52, find the vector \(\mathbf{v}\) with the given magnitude and the same direction as \(\mathbf{u}\). Magnitude - ||\(\mathbf{v}\)|| \(= 9\) Direction - \(\mathbf{u} = \langle 2, 5 \rangle\)

Step-by-Step Solution

Answer

The normalized direction vector \(\mathbf{u}_n\) is \(\mathbf{u}_n = \mathbf{u} / \|\mathbf{u}\|\). The vector \(\mathbf{v}\) with the given magnitude and the same direction as \(\mathbf{u}\) is found by calculating \(\mathbf{v} = \mathbf{u}_n \times 9\).

Step 1: Normalize the Direction Vector

The direction vector \(\mathbf{u}\) needs to be normalized first. The normalized vector is found by dividing each component of the vector by the magnitude of the vector. The magnitude of a vector is found using the formula \(\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2}\). So, the magnitude of \(\mathbf{u}\) would be \(\|\mathbf{u}\| = \sqrt{2^2 + 5^2}\). Thus, the normalized vector \(\mathbf{u}_n\) is \(\mathbf{u}_n = \mathbf{u} / \|\mathbf{u}\|\).

Step 2: Multiply the Normalized Direction Vector by the Desired Magnitude

To get the vector \(\mathbf{v}\) which has the same direction as \(\mathbf{u}\) and a desired magnitude, the normalized vector \(\mathbf{u}_n\) found in step 1 is multiplied by the magnitude ||\(\mathbf{v}\)|| = 9. This will extend or compress the length of the vector. So, the desired vector \(\mathbf{v}\) will be \(\mathbf{v} = \mathbf{u}_n \times 9\).

Step 3: Calculate the components of \(\mathbf{v}\)

In this step, the components of the vector \(\mathbf{v}\) are calculated by multiplying the components of the normalized vector \(\mathbf{u}_n\) by the magnitude ||\(\mathbf{v}\)|| = 9.

Key Concepts

Magnitude of a Vector

Understanding the magnitude of a vector is fundamental in physics and mathematics, as it represents the length or size of the vector. In essence, the magnitude is a measure of how far the vector extends in space. It is calculated using the Pythagorean theorem for a vector with components \(x, y\) in two dimensions, or \(x, y, z\) in three dimensions.

For a two-dimensional vector \(\mathbf{u} = \langle u_1, u_2 \rangle\), the magnitude, denoted by \(\|\mathbf{u}\|\), is found using the formula: \[\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2}\]. In a practical scenario, if you have a direction vector with components \(2, 5\), the magnitude is \(\|\mathbf{u}\| = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}\), which is approximately 5.39.

When solving problems, it's essential to calculate the magnitude accurately, as it affects subsequent computations, such as when normalizing vectors or determining vector directions.

Direction Vector

The direction of a vector is just as crucial as its magnitude. It describes the orientation of the vector in space and can be depicted by an arrow pointing from its point of origin to its end. To work with the direction in a more quantifiable way, we use a 'direction vector'. A direction vector is often provided in its raw form and needs to be normalized.

Normalizing a vector involves dividing each of its components by its magnitude to produce a unit vector. This unit vector maintains the original vector's direction but has a magnitude of 1. For example, the two-dimensional vector \(\mathbf{u} = \langle 2, 5 \rangle\) is normalized as follows: Firstly, calculate its magnitude (as described in the prior section), and secondly, divide each component by this magnitude to achieve the normalized vector \(\mathbf{u}_n\).

In practice, this process of normalization allows us to easily scale vectors to any desired length while keeping their direction intact. It is particularly handy when needing to calculate forces, velocities, or other vector quantities in physics.

Vector Components

Vector components are a powerful way to dissect vectors into manageable parts, usually corresponding to the dimensions of the space they exist in. For a vector situated in a two-dimensional plane, the components are its x (horizontal) and y (vertical) parts. In three dimensions, a z component is added, representing the depth.

These components are fundamental as they allow vectors to be analyzed and manipulated mathematically. For operations such as adding vectors, subtracting them, or finding their dot product, the individual components become incredibly useful. For instance, if a vector \(\mathbf{v}\) has to have the same direction as a vector \(\mathbf{u}\) but with a different magnitude, you will first normalize \(\mathbf{u}\), then multiply its components by the new magnitude to derive the components of \(\mathbf{v}\).

Through the lens of our example, the normalized vector components of \(\mathbf{u}\) are found, and each is then multiplied by the new magnitude, \(9\), to acquire the desired vector \(\mathbf{v}\). This illustrates how the individual vector components are used to adjust a vector’s magnitude while preserving its direction.