Problem 77 - Precalculus with Calculus Previews

Question

Find a unit vector in the opposite direction of \(\mathbf{v}=\) \(\langle 10,-5,10\rangle .\)

Step-by-Step Solution

Answer

The unit vector in the opposite direction is \( \langle -\frac{2}{3}, \frac{1}{3}, -\frac{2}{3} \rangle.\)

Step 1: Calculate the Magnitude of the Vector

First, we need to calculate the magnitude of the vector \( \mathbf{v} = \langle 10, -5, 10 \rangle \). The magnitude is given by the formula: \[\| \mathbf{v} \| = \sqrt{10^2 + (-5)^2 + 10^2} = \sqrt{100 + 25 + 100} = \sqrt{225} = 15.\]Thus, the magnitude of \( \mathbf{v} \) is 15.

Step 2: Find the Unit Vector in the Opposite Direction

To find the unit vector in the opposite direction of \( \mathbf{v} \), first, find the unit vector in the direction of \( \mathbf{v} \) by dividing each component by the magnitude calculated:\[\frac{1}{\|\mathbf{v}\|} \mathbf{v} = \frac{1}{15} \langle 10, -5, 10 \rangle = \langle \frac{2}{3}, -\frac{1}{3}, \frac{2}{3} \rangle.\]Then, change the direction by multiplying each component by -1:\[-\left(\langle \frac{2}{3}, -\frac{1}{3}, \frac{2}{3} \rangle\right) = \langle -\frac{2}{3}, \frac{1}{3}, -\frac{2}{3} \rangle.\]

Key Concepts

Vector Magnitude

The magnitude of a vector is crucial in understanding its size or length. To calculate the magnitude of a vector, we apply the distance formula. For a 3-dimensional vector such as \(\mathbf{v} = \langle x, y, z \rangle\), the magnitude \(\|\mathbf{v}\|\) is given by:

• \(\|\mathbf{v}\| = \sqrt{x^2 + y^2 + z^2} \).

This formula is derived from the Pythagorean theorem, and it provides a measurement of the vector from the origin in a Euclidean space. In our given problem, the vector \(\langle 10, -5, 10 \rangle\) has a magnitude calculated as \(\sqrt{10^2 + (-5)^2 + 10^2} = \sqrt{225} = 15\).
Knowing the magnitude helps us to normalize the vector when finding unit vectors, as well as indicating the physical quantities in physics such as speed and force.

Opposite Direction

Finding a vector in the opposite direction is a concept used when you need to reverse the path of a vector. This is important in navigation and engineering. To reverse a vector, you simply multiply each component by \(-1\). It involves the following steps:

• Identify the unit vector: First, you find the unit vector in the same direction by dividing each vector component by the vector's magnitude.
• Change the direction: Then, multiply each component of the unit vector by \(-1\).

In our exercise, after finding the unit vector \(\langle \frac{2}{3}, -\frac{1}{3}, \frac{2}{3}\rangle\), we multiply through by \(-1\) to get the opposite direction: \(\langle -\frac{2}{3}, \frac{1}{3}, -\frac{2}{3}\rangle\).
This inverse explains how objects or forces might interact in opposite trajectories or how movements are described in reverse.

Vector Scaling

Vector scaling refers to adjusting the magnitude of a vector without altering its direction. A fundamental vector operation, scaling is done by multiplying the vector by a scalar, which is a real number. Here's how it applies:

• Scaling up: If the scalar is greater than 1, the vector’s length increases proportionally.
• Scaling down: If the scalar is less than 1, it reduces the vector's length. With a scalar of zero, the vector collapses to the zero vector.

In the context of unit vectors, scaling is used to generate a vector of magnitude 1, necessary for normalized vectors. For example, in our problem, to find the unit vector of \(\mathbf{v} = \langle 10, -5, 10 \rangle\), each component was divided by its magnitude of 15. This operation resized \(\mathbf{v}\) to \(\langle \frac{2}{3}, -\frac{1}{3}, \frac{2}{3} \rangle\),
allowing us to maintain directionality while adjusting the magnitude to meet specific requirements in applications such as physics and computer graphics.