The vectors **u** and **v** have the same direction. 

**a.** Find 

left norm Bold u right norm**u**.

**b.** Find 

left norm Bold v right norm**v**.

**c.** Is 

**u**equals=**v**​? 

Explain. **.** 

left norm Bold u right norm**u**equals=enter your response here 

​(Simplify your answer. Type an exact​ answer, using radicals as​ needed.)

Let the vectors **u** and **v** have the same direction. Without explicit details about their components, we'll work with the general case to solve the problem, interpreting this symbolically.

### a. Find \( \lVert \mathbf{u} \rVert \mathbf{u} \)
The expression \( \lVert \mathbf{u} \rVert \mathbf{u} \) involves multiplying the magnitude (or norm) of the vector **u** by the vector **u** itself. Since the norm (or magnitude) of a vector is a scalar:

\[
\lVert \mathbf{u} \rVert \mathbf{u} = \lVert \mathbf{u} \rVert \cdot \mathbf{u}
\]

This gives us the vector **u** scaled by its magnitude.

### b. Find \( \lVert \mathbf{v} \rVert \mathbf{v} \)
Similarly, \( \lVert \mathbf{v} \rVert \mathbf{v} \) is the norm of vector **v** multiplied by the vector **v** itself:

\[
\lVert \mathbf{v} \rVert \mathbf{v} = \lVert \mathbf{v} \rVert \cdot \mathbf{v}
\]

This is the vector **v** scaled by its magnitude.

### c. Is \( \mathbf{u} = \mathbf{v} \)? Explain.
Vectors **u** and **v** have the same direction, meaning that they are scalar multiples of each other. In other words:

\[
\mathbf{u} = c \cdot \mathbf{v}
\]

for some scalar \( c \). For \( \mathbf{u} \) and \( \mathbf{v} \) to be equal, the scalar \( c \) must be 1 (i.e., they must have the same magnitude as well as the same direction).

If the magnitudes of **u** and **v** are different, then \( \mathbf{u} \neq \mathbf{v} \). If the magnitudes are the same, then \( \mathbf{u} = \mathbf{v} \).

### Left norm \( \mathbf{u} \) equals

Given that the vectors **u** and **v** have the same direction, we conclude that:

\[
\lVert \mathbf{u} \rVert = \lVert \mathbf{v} \rVert
\]

Thus, the simplified form of \( \lVert \mathbf{u} \rVert \mathbf{u} \) is just:

\[
\lVert \mathbf{u} \rVert^2 \cdot \mathbf{v}
\]

(Since they are in the same direction and we are assuming the magnitudes to be equal).

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Would you like further details on any part of the explanation?

Here are 5 related questions:
1. How do you find the magnitude of a vector?
2. How do vectors differ when they have opposite directions but the same magnitude?
3. Can two vectors have the same magnitude but different directions? What does that imply?
4. What happens if two vectors have the same direction but different magnitudes?
5. How do you represent a vector as a unit vector?

**Tip:** To determine if two vectors are equal, always check both their magnitude and direction.

The vectors **u** and **v** have the same direction.

a. Find \( \lVert \mathbf{u} \rVert \mathbf{u} \).

b. Find \( \lVert \mathbf{v} \rVert \mathbf{v} \).

c. Is **u** equal to **v**? Explain.

Explore how to determine if vectors u and v, which share the same direction, are equal by using vector norms and scalar multiplication. This detailed explanation covers vector norm calculations and the conditions for vector equality, suitable for high school students in Grades 9-12.

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