Problem 50 - 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 - ||\(\mathbf{v}\)|| \(= 3\) Direction - \(\mathbf{u} = \langle -12, -5 \rangle\)

Step-by-Step Solution

Answer

The vector \(\mathbf{v}\) with the given magnitude and the same direction as \(\mathbf{u}\) is \(\langle \frac{-36}{13}, \frac{-15}{13} \rangle\).

Step 1: Calculate the Magnitude of the Given Vector

First, calculate the magnitude of the vector \(\mathbf{u} = \langle -12, -5 \rangle\). The magnitude of a vector \(\mathbf{u} = \langle a, b \rangle\) is given by \(\|\mathbf{u}\| = \sqrt{a^2 + b^2}\). Thus, you get \(\|\mathbf{u}\| = \sqrt{(-12)^2 + (-5)^2} = 13\).

Step 2: Find the Unit Vector of u

The unit vector of \(\mathbf{u}\) (let's denote it with \(\mathbf{\hat{u}}\)) is the vector that has the same direction as \(\mathbf{u}\) but a magnitude of 1. To obtain the unit vector, divide each component of the original vector by its magnitude, which gives \(\mathbf{\hat{u}} = \langle \frac{-12}{13}, \frac{-5}{13} \rangle\).

Step 3: Find vector v with the given magnitude and the same direction as u

To find the vector \(\mathbf{v}\) with the given magnitude and the same direction as \(\mathbf{u}\), multiply the unit vector \(\mathbf{\hat{u}}\) by the target magnitude ||\(\mathbf{v}\)|| \(= 3\). Therefore, \(\mathbf{v} = 3 \cdot \langle \frac{-12}{13}, \frac{-5}{13} \rangle = \langle \frac{-36}{13}, \frac{-15}{13} \rangle\).

Key Concepts

Magnitude of a Vector

When we talk about the magnitude of a vector, we are referring to its "length" or "size". Imagine a vector as an arrow drawn on a graph that points from one location to another. The magnitude is how long that arrow is, independent of its direction.
To calculate the magnitude of a vector given its components, say \( \mathbf{u} = \langle a, b \rangle \), you use the formula:

• \( \| \mathbf{u} \| = \sqrt{a^2 + b^2} \)

The square root is needed because you're applying the Pythagorean theorem, converting the lengths of the vertical and horizontal components into a single diagonal length, namely the magnitude. For example, with \( \mathbf{u} = \langle -12, -5 \rangle \), the magnitude is calculated with \( \| \mathbf{u} \| = \sqrt{(-12)^2 + (-5)^2} = 13 \).
Knowing the magnitude helps in various calculations, especially when you want to work with vectors affecting distances, like applying scales or understanding force directions in physics.

Unit Vector

A unit vector is a vector that has a magnitude of exactly one unit, while still maintaining its direction. Think of it as a way to convey only direction, without considering how far you are traveling. This is particularly useful when we want to change the magnitude of another vector but keep the direction intact, or vice versa.

• To find a unit vector, divide each component of the original vector by its magnitude.
• For instance, given the vector \( \mathbf{u} = \langle -12, -5 \rangle \) with magnitude \( 13 \), the unit vector \( \mathbf{\hat{u}} \) is obtained as \( \mathbf{\hat{u}} = \langle \frac{-12}{13}, \frac{-5}{13} \rangle \).

This process normalizes the vector, scaling it down to a length of one. The resultant unit vector serves as a direction indicator, while a scalar multiplication can easily convert it back to a desired magnitude.

Direction of a Vector

The direction of a vector refers to the way in which the vector points in the coordinate space. Even though vectors have both direction and magnitude, in some cases, emphasis on direction is needed, especially when manipulating vectors, like scaling them while preserving their orientation.
Direction is intrinsically linked with unit vectors. Any vector, regardless of its size, can point in a particular direction, but when you express it as a unit vector, the direction becomes the focal point. It's as if you "strip" away the magnitude to just focus on which way it's going.
When determining the direction, using unit vectors simplifies the task. Consider the vector \( \mathbf{u} = \langle -12, -5 \rangle \). Even if \( \mathbf{u} \) were scaled to a different magnitude, say, to length 3 to form \( \mathbf{v} = \langle \frac{-36}{13}, \frac{-15}{13} \rangle \), it still points in the same direction as the original vector \( \mathbf{u} \).
This property is vital for applications like navigation, animation paths in gaming, and more, as it allows consistent translation and transformation of vector paths without altering their intended trajectory.