https://file.aiquickdraw.com/mathbot/file/1730300760897iheytfeq.jpg

To solve this problem, let’s go through it step-by-step.

### 1. Finding the Length of Vector **a**
The vector **a** is given as:
\[
\mathbf{a} = \begin{bmatrix} 6 \\ 3 \\ 1 \end{bmatrix}
\]
The length (or magnitude) of a vector **a** with components \( a_x \), \( a_y \), and \( a_z \) is calculated using the formula:
\[
\|\mathbf{a}\| = \sqrt{a_x^2 + a_y^2 + a_z^2}
\]
For this vector:
\[
\|\mathbf{a}\| = \sqrt{6^2 + 3^2 + 1^2} = \sqrt{36 + 9 + 1} = \sqrt{46}
\]
Thus, the length of **a** is:
\[
\|\mathbf{a}\| = \sqrt{46} \approx 6.78
\]

### 2. Finding the Unit Vector in the Same Direction as **a**
The unit vector in the direction of **a** is obtained by dividing each component of **a** by the magnitude \(\|\mathbf{a}\|\). The unit vector **u** in the direction of **a** is given by:
\[
\mathbf{u} = \frac{1}{\|\mathbf{a}\|} \mathbf{a} = \frac{1}{\sqrt{46}} \begin{bmatrix} 6 \\ 3 \\ 1 \end{bmatrix}
\]
Breaking this down, each component of the unit vector is:
\[
u_x = \frac{6}{\sqrt{46}}, \quad u_y = \frac{3}{\sqrt{46}}, \quad u_z = \frac{1}{\sqrt{46}}
\]
If we approximate these values:
\[
u_x \approx 0.884, \quad u_y \approx 0.442, \quad u_z \approx 0.147
\]
Thus, the unit vector **u** is:
\[
\mathbf{u} \approx \begin{bmatrix} 0.884 \\ 0.442 \\ 0.147 \end{bmatrix}
\]

### Summary of Answers
- **Length of \( \mathbf{a} \)**: \( \sqrt{46} \approx 6.78 \)
- **Unit Vector in the Direction of \( \mathbf{a} \)**: \( \begin{bmatrix} 0.884 \\ 0.442 \\ 0.147 \end{bmatrix} \)

Would you like more detailed steps, or do you have any questions about the solution?

---

### Related Questions
1. How do you find the magnitude of a 2-dimensional vector?
2. What is the purpose of finding a unit vector?
3. Can a unit vector have a negative component?
4. How is vector addition performed?
5. What are the applications of vectors in physics?

### Tip
Always double-check your square root calculations, as rounding errors can impact the precision of your answers, especially when finding unit vectors.

For the vector a = [6, 3, 1], find:
1. The length of a
2. The unit vector in the same direction as a

Calculate the length and unit vector for the vector a = [6, 3, 1]. Learn to find the magnitude of a vector and determine the unit vector in its direction, suitable for high school 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