Find the magnitude 

left norm Bold v right norm**v**​, 

to the nearest​ hundredth, and the direction 

thetaθ​, 

to the nearest tenth of a​ degree, for the given vector 

**v**.

Bold v equals left parenthesis 6 Bold i minus 5 Bold j right parenthesis minus left parenthesis 6 Bold i minus 8 Bold j right parenthesis**v**=(6**i**−5**j**)−(6**i**−8**j**)

Question content area bottom

Part 1

left norm Bold v right norm**v** 

​= 

enter your response here  

​(Round to the nearest hundredth as​ needed.)

We are given the vector **v** and we are tasked with finding its magnitude and direction. Let's break this problem down step by step.

### Step 1: Simplify the given vector expression

The vector **v** is given as:

\[
\mathbf{v} = (6 \mathbf{i} - 5 \mathbf{j}) - (6 \mathbf{i} - 8 \mathbf{j})
\]

We simplify by distributing the minus sign:

\[
\mathbf{v} = (6 \mathbf{i} - 5 \mathbf{j}) - 6 \mathbf{i} + 8 \mathbf{j}
\]

Now, combine like terms:

\[
\mathbf{v} = (6 \mathbf{i} - 6 \mathbf{i}) + (-5 \mathbf{j} + 8 \mathbf{j}) = 0 \mathbf{i} + 3 \mathbf{j}
\]

Thus, the simplified vector is:

\[
\mathbf{v} = 3 \mathbf{j}
\]

### Step 2: Find the magnitude of the vector

The magnitude of a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} \) is calculated as:

\[
\|\mathbf{v}\| = \sqrt{a^2 + b^2}
\]

For our vector \( \mathbf{v} = 0 \mathbf{i} + 3 \mathbf{j} \), we have \( a = 0 \) and \( b = 3 \). Thus, the magnitude is:

\[
\|\mathbf{v}\| = \sqrt{0^2 + 3^2} = \sqrt{9} = 3
\]

### Step 3: Find the direction (θ)

The direction of the vector is given by:

\[
\theta = \tan^{-1} \left( \frac{b}{a} \right)
\]

Since \( a = 0 \) and \( b = 3 \), the direction is straight along the positive \( j \)-axis (90 degrees):

\[
\theta = \tan^{-1} \left( \frac{3}{0} \right) = 90^\circ
\]

### Final Answers:
- Magnitude: \( 3 \)
- Direction: \( 90^\circ \)

Let me know if you need further details or clarifications.

---

Here are 5 related questions for practice:

1. What is the magnitude of the vector \( \mathbf{v} = 4 \mathbf{i} + 3 \mathbf{j} \)?
2. Find the direction of the vector \( \mathbf{v} = -3 \mathbf{i} + 4 \mathbf{j} \).
3. How do you compute the magnitude of a vector in 3D space?
4. What is the direction of a vector lying along the negative \( j \)-axis?
5. If \( \mathbf{v} = 5 \mathbf{i} + 12 \mathbf{j} \), what are its magnitude and direction?

**Tip:** When finding the direction of a vector, always be cautious of quadrant positioning when using the inverse tangent function.

Find the magnitude and direction θ, to the nearest hundredth and tenth of a degree, respectively, for the given vector **v** = (6i - 5j) - (6i - 8j).

Learn how to find the magnitude and direction of the vector v = (6i - 5j) - (6i - 8j). This problem involves simplifying vectors, calculating magnitudes, and determining direction using trigonometric functions. 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