Consider the vector ~v = h1, 1i in the plane (i.e. R2 ). Find all unit vectors which are normal to ~v.

Given the vector \(\mathbf{v} = \langle 1, 1 \rangle\) in \(\mathbb{R}^2\), we are tasked with finding all **unit vectors** that are **normal** (i.e., perpendicular) to \(\mathbf{v}\).

### Step 1: Condition for perpendicularity
A vector \(\mathbf{u} = \langle u_1, u_2 \rangle\) is **normal** to \(\mathbf{v}\) if their dot product is zero. The dot product of two vectors \(\mathbf{v} = \langle 1, 1 \rangle\) and \(\mathbf{u} = \langle u_1, u_2 \rangle\) is:

\[
\mathbf{v} \cdot \mathbf{u} = 1 \cdot u_1 + 1 \cdot u_2 = u_1 + u_2
\]

For \(\mathbf{u}\) to be perpendicular to \(\mathbf{v}\), we set the dot product to zero:

\[
u_1 + u_2 = 0 \quad \text{or} \quad u_1 = -u_2
\]

### Step 2: Condition for unit vectors
A unit vector \(\mathbf{u} = \langle u_1, u_2 \rangle\) must also satisfy the condition that its magnitude is 1. The magnitude of \(\mathbf{u}\) is:

\[
|\mathbf{u}| = \sqrt{u_1^2 + u_2^2}
\]

Setting this equal to 1:

\[
\sqrt{u_1^2 + u_2^2} = 1 \quad \Rightarrow \quad u_1^2 + u_2^2 = 1
\]

### Step 3: Solve the system
From the perpendicularity condition, we know \(u_1 = -u_2\). Substituting this into the unit vector condition:

\[
u_1^2 + (-u_1)^2 = 1 \quad \Rightarrow \quad 2u_1^2 = 1 \quad \Rightarrow \quad u_1^2 = \frac{1}{2} \quad \Rightarrow \quad u_1 = \pm \frac{1}{\sqrt{2}}
\]

Thus, \(u_2 = -u_1\), giving us:

- If \(u_1 = \frac{1}{\sqrt{2}}\), then \(u_2 = -\frac{1}{\sqrt{2}}\)
- If \(u_1 = -\frac{1}{\sqrt{2}}\), then \(u_2 = \frac{1}{\sqrt{2}}\)

### Step 4: Final answer
The two unit vectors that are perpendicular to \(\mathbf{v} = \langle 1, 1 \rangle\) are:

\[
\mathbf{u}_1 = \left\langle \frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}} \right\rangle \quad \text{and} \quad \mathbf{u}_2 = \left\langle -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right\rangle
\]

These are the two unit vectors that are normal to \(\mathbf{v}\).

---

Do you want further details, or do you have any other questions?

Here are five related questions you might consider:
1. What is the geometric interpretation of perpendicular vectors in \(\mathbb{R}^2\)?
2. How can we generalize this process to find normal vectors in \(\mathbb{R}^3\)?
3. How can we calculate the angle between two vectors in \(\mathbb{R}^2\)?
4. What is the importance of unit vectors in vector operations and physics?
5. How do we define a normal vector in higher-dimensional spaces?

**Tip**: When working with vectors, always check for both magnitude and direction conditions when solving for unit or perpendicular vectors.

Consider the vector ~v = h1, 1i in the plane (i.e. R^2). Find all unit vectors which are normal to ~v.

This solution details how to find all unit vectors that are perpendicular to the vector (1,1) in R^2. The process involves using the dot product and magnitude formulas to solve for perpendicular unit vectors. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation