Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.$$\mathbf{v}=\langle-1,1\rangle$$.

Step-by-Step Solution

Answer

The unit vector in the direction of the vector \(\mathbf{v}=\langle-1,1\rangle\) is \(\mathbf{u}=\langle-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\rangle\).

Step 1: Compute the Magnitude (Norm) of Vector

We first compute the magnitude of the vector v using the formula for the magnitude of two-dimensional vectors, which is the square-root of the sum of the squares of the components of v. So, magnitude of \(\mathbf{v}\) = \(||\mathbf{v}|| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}\)

Step 2: Compute the Unit Vector

To find the unit vector \(\mathbf{u}\) in the direction of \(\mathbf{v}\), we divide the vector \(\mathbf{v}\) by its magnitude. So, \(\mathbf{u} = \dfrac{\mathbf{v}}{||\mathbf{v}||} = \dfrac{1}{\sqrt{2}}\langle-1, 1\rangle=\langle-\dfrac{1}{\sqrt{2}}, \dfrac{1}{\sqrt{2}}\rangle\).

Step 3: Verification

We can verify that the result is indeed a unit vector by ensuring that its magnitude is 1. The magnitude of the calculated unit vector \(\mathbf{u}\) = \(\sqrt{(-\frac{1}{\sqrt{2}})^2 + (\frac{1}{\sqrt{2}})^2} = 1\). As the magnitude of \(\mathbf{u}\) equals 1, \(\mathbf{u}\) is indeed a unit vector

Key Concepts

Understanding Vector Magnitude

The magnitude of a vector is a crucial concept in understanding its size or length. For a two-dimensional vector, such as \(\mathbf{v} = \langle -1, 1 \rangle\), the magnitude is calculated using the Pythagorean theorem.

The formula to find the magnitude \(||\mathbf{v}||\) of a vector \(\langle a, b \rangle\) is:

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

This formula essentially sums the squares of the vector's components and takes the square root.

For \(\mathbf{v} = \langle -1, 1 \rangle\), it's calculated as:

• \(||\mathbf{v}|| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}\)

This tells us the vector's total length in the plane.

Exploring Two-Dimensional Vectors

Two-dimensional vectors are ordered pairs, typically represented as \(\langle x, y \rangle\). They describe direction and magnitude in a flat, 2D space. Understanding these vectors helps solve many practical physics and math problems.

Each component represents direction along the x- and y-axes. For example, \(\mathbf{v} = \langle -1, 1 \rangle\) moves left by 1 and up by 1.

• Horizontal movement: Determined by the x-component.
• Vertical movement: Determined by the y-component.

Such vectors can define positions, forces, and velocities, making them versatile in various applications.

Unit Vector Verification Process

A unit vector has a magnitude of exactly 1. It retains the direction of the original vector but scales down its length. To find a unit vector \(\mathbf{u}\) from a given vector \(\mathbf{v}\), you divide each component of \(\mathbf{v}\) by its magnitude.

Here's how to verify:

• Calculate: \(\mathbf{u} = \dfrac{1}{||\mathbf{v}||} \langle -1, 1 \rangle = \langle -\dfrac{1}{\sqrt{2}}, \dfrac{1}{\sqrt{2}} \rangle\)
• Check magnitude: \(||\mathbf{u}|| = \sqrt{\left(-\dfrac{1}{\sqrt{2}}\right)^2 + \left(\dfrac{1}{\sqrt{2}}\right)^2} = 1\)

The unit vector \(\mathbf{u}\) is confirmed by its magnitude of 1, proving that it is indeed a unit vector.