Problem 52 - Precalculus with Limits

Question

Finding a Vector In Exercises \(49-52,\) find the vector \(v\) with the given magnitude and the same direction as u. \(\begin{array}{ll}{\text { Magnitude }} & {\text { Direction }} \\\ {\|\mathbf{y}\|=8} & {\mathbf{y}=\langle 3,3\rangle}\end{array}\)

Step-by-Step Solution

Answer

The vector \(v\) with the same direction as \(\langle 3,3 \rangle\) and a magnitude of 8 is \(\langle 8\frac{3}{\sqrt{18}}, 8\frac{3}{\sqrt{18}} \rangle\).

Step 1: Find the magnitude of the given vector

First, find the magnitude of the given vector u (\(u = \langle 3,3 \rangle\)). The formula for the magnitude of a 2D vector is \(\|u\| = \sqrt{x^2 + y^2}\), where \(x\) and \(y\) are the coordinates of the vector. Now replace \(x\) and \(y\) with the coordinates 3 and 3: \(\|u\| = \sqrt{3^2 + 3^2} = \sqrt{18}\).

Step 2: Find the unit vector

Now, find the unit vector of \(u\) by dividing the vector \(u\) by its magnitude. The unit vector \(u_{unit}\) is \(\frac{u}{\|u\|}\). In this case \(u_{unit} = \frac{\langle 3,3 \rangle}{\sqrt{18}} = \langle \frac{3}{\sqrt{18}}, \frac{3}{\sqrt{18}} \rangle\). This gives the unit vector in the same direction as \(u\).

Step 3: Calculate the vector v

Now that the unit vector is found, \(v\) can be found by multiplying the unit vector by the desired magnitude of \(v\), which is 8: \(v = \|v\| \cdot u_{unit} = 8 \cdot \langle \frac{3}{\sqrt{18}}, \frac{3}{\sqrt{18}} \rangle = \langle 8\frac{3}{\sqrt{18}}, 8\frac{3}{\sqrt{18}} \rangle\). This is the vector \(v\) with a magnitude of 8 and the same direction as \(u\).

Key Concepts

Magnitude of a Vector

The magnitude of a vector is like the length of an arrow from its starting point to its end point. In mathematical terms, it's the size of the vector. To find the magnitude of a 2D vector, which is a vector with two components, we use the formula:

• \( \|u\| = \sqrt{x^2 + y^2} \)

Here, \(x\) and \(y\) are the components of the vector. For example, if we have a vector \(u = \langle 3, 3 \rangle \), we plug these values into the formula. So, \( \|u\| = \sqrt{3^2 + 3^2} = \sqrt{18} \), which simplifies to approximately 4.24.
This measurement essentially helps in understanding how "long" or "intense" the vector is without considering its direction. It's an important part of vector mathematics and serves as a foundation for understanding other concepts, like unit vectors and vector direction.

Unit Vector

A unit vector is a special vector that has a magnitude of exactly 1. In other words, it retains the direction of the original vector but has a standardized length. Creating a unit vector involves dividing the original vector by its magnitude, thereby normalizing it:

• \( u_{unit} = \frac{u}{\|u\|} \)

This calculation helps us isolate the vector’s direction, which can be useful for scaling the vector to a desired length without changing its direction. For example, if your vector \(u\) is \(\langle 3, 3 \rangle\) and its magnitude is \(\sqrt{18}\), the unit vector \(u_{unit}\) would be \(\langle \frac{3}{\sqrt{18}}, \frac{3}{\sqrt{18}} \rangle\).
Unit vectors are crucial for operations that require standard direction vectors, essentially acting as building blocks for more complex vector calculations.

Direction of a Vector

The direction of a vector is tied to the path it points along. In a 2D space, this direction can be expressed as an angle from the positive x-axis or simply by the vector's slope—a reflection of how it moves in space.
For the vector \( \langle 3, 3 \rangle \), the direction can be glimpsed through equal components, suggesting a 45-degree angle from the positive x-axis since every unit of movement in the x-direction matches the y-direction. In scenarios involving more complex vectors, direction is often calculated using trigonometry, where:

• The direction angle \( \theta \) is found using \( \tan(\theta) = \frac{y}{x} \), where \(x\) and \(y\) are vector coordinates.

Understanding the direction is especially important in applications like physics and computer graphics where the orientation of force or movement matters extensively.

2D Vector Calculation

Calculating a 2D vector involves several steps to understanding both its physical characteristics and applications. Primarily, you need to grasp the magnitude, unit vectors, and direction as discussed.
In practical scenarios, like transforming vector \(u = \langle 3, 3 \rangle \) to another vector with the same direction but a different magnitude (like a size of 8), involves integrating these concepts. You first extract the unit vector from \(u\), then scale it by the desired magnitude.

• New Vector \( v = 8 \cdot u_{unit} \)
• \( v = 8 \cdot \langle \frac{3}{\sqrt{18}}, \frac{3}{\sqrt{18}} \rangle \)
• \( v = \langle 8 \frac{3}{\sqrt{18}}, 8 \frac{3}{\sqrt{18}} \rangle \)

These steps encapsulate what vector mathematics achieve: manipulating vector quantities in both direction and magnitude to serve specific purposes, whether in statics, dynamics, or any engineering applications.