https://file.aiquickdraw.com/mathbot/file/1730996178618s9asmbip.jpg

To solve this problem, we need to follow these steps:

### Step 1: Define the Given Vectors
The given vectors are:
\[
\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}
\]
\[
\vec{b} = -\hat{i} + \hat{j} - \hat{k}
\]

### Step 2: Find \(\vec{a} + \vec{b}\)
Add the components of \(\vec{a}\) and \(\vec{b}\) to find the resultant vector \(\vec{a} + \vec{b}\):
\[
\vec{a} + \vec{b} = (2 - 1)\hat{i} + (-1 + 1)\hat{j} + (2 - 1)\hat{k}
\]
Simplifying each component:
\[
\vec{a} + \vec{b} = \hat{i} + 0\hat{j} + \hat{k}
\]
So,
\[
\vec{a} + \vec{b} = \hat{i} + \hat{k}
\]

### Step 3: Find the Magnitude of \(\vec{a} + \vec{b}\)
The magnitude of \(\vec{a} + \vec{b}\) is calculated as follows:
\[
|\vec{a} + \vec{b}| = \sqrt{(1)^2 + (0)^2 + (1)^2}
\]
\[
|\vec{a} + \vec{b}| = \sqrt{1 + 0 + 1} = \sqrt{2}
\]

### Step 4: Find the Unit Vector in the Direction of \(\vec{a} + \vec{b}\)
To find the unit vector in the direction of \(\vec{a} + \vec{b}\), divide each component of \(\vec{a} + \vec{b}\) by its magnitude:
\[
\text{Unit vector} = \frac{\vec{a} + \vec{b}}{|\vec{a} + \vec{b}|} = \frac{\hat{i} + \hat{k}}{\sqrt{2}}
\]
Thus, the unit vector is:
\[
\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{k} = \frac{\hat{i} + \hat{k}}{\sqrt{2}}
\]

### Final Answer
The unit vector in the direction of \(\vec{a} + \vec{b}\) is:
\[
\frac{\hat{i} + \hat{k}}{\sqrt{2}}
\]

Would you like further clarification on any steps, or do you have more questions?

### Additional Questions
1. How would we calculate the unit vector if the magnitudes had complex numbers?
2. Can you explain why we use the magnitude to find the unit vector?
3. What if \(\vec{a}\) and \(\vec{b}\) had more dimensions, like in 4D space?
4. How can we use this method for unit vectors in physics applications?
5. Is there an alternative approach to find the unit vector?

### Tip
Always ensure the magnitude is positive when finding unit vectors, as it represents the scalar length of a vector.

For given vectors \( \vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} \) and \( \vec{b} = -\hat{i} + \hat{j} - \hat{k} \), find the unit vector in the direction of the vector \( \vec{a} + \vec{b} \).

Learn how to find the unit vector in the direction of the vector sum \( \vec{a} + \vec{b} \) for given vectors \( \vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} \) and \( \vec{b} = -\hat{i} + \hat{j} - \hat{k} \). This problem explores vector addition, magnitude, and 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