Let W be the union of the 

firstfirst 

and 

thirdthird 

quadrants in the​ xy-plane. That​ is, let 

Upper W equals StartSet Start 2 By 1 Table 1st Row 1st Column x 2nd Row 1st Column y EndTable : xy greater than or equals 0 EndSetW=

x

y

 : xy≥0. 

Complete parts a and b below.

Question content area bottom

Part 1

a. If **u** is in W and c is any​ scalar, is 

c**u**

 in​ W? Why?

A.

If 

**u**equals=Start 2 By 1 Table 1st Row 1st Column x 2nd Row 1st Column y EndTable

x

y
 
is in​ W, then the vector 

c**u**equals=cStart 2 By 1 Table 1st Row 1st Column x 2nd Row 1st Column y EndTable

x

y

equals=Start 2 By 1 Table 1st Row 1st Column cx 2nd Row 1st Column cy EndTable

cx

cy
 
is in W because 

left parenthesis cx right parenthesis left parenthesis cy right parenthesis(cx)(cy)equals=c squared left parenthesis xy right parenthesis greater than or equals 0c2(xy)≥0 

since 

xy greater than or equals 0xy≥0.

B.

If 

**u**equals=Start 2 By 1 Table 1st Row 1st Column x 2nd Row 1st Column y EndTable

x

y
 
is in​ W, then the vector 

c**u**equals=cStart 2 By 1 Table 1st Row 1st Column x 2nd Row 1st Column y EndTable

x

y

equals=Start 2 By 1 Table 1st Row 1st Column cx 2nd Row 1st Column cy EndTable

cx

cy
 
is not in W because 

cxxyyless than or equals≤0 

in some cases.

C.

If 

**u**equals=Start 2 By 1 Table 1st Row 1st Column x 2nd Row 1st Column y EndTable

x

y
 
is in​ W, then the vector 

c**u**equals=cStart 2 By 1 Table 1st Row 1st Column x 2nd Row 1st Column y EndTable

x

y

equals=Start 2 By 1 Table 1st Row 1st Column cx 2nd Row 1st Column cy EndTable

cx

cy
 
is in W because 

cxxyygreater than or equals≥0 

since 

xy greater than or equals 0xy≥0.

Let's analyze the question.

### Given:
- The set \( W \) is defined as the union of the first and third quadrants in the \( xy \)-plane, i.e., \( W = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : xy \geq 0 \right\} \).
- \( u = \begin{pmatrix} x \\ y \end{pmatrix} \) is a vector in \( W \).
- We need to determine if \( c \cdot u \), where \( c \) is any scalar, is also in \( W \), and we need to choose the correct option from A, B, or C.

### Analysis:
- For \( u \) to be in \( W \), the product \( xy \) must be non-negative (\( xy \geq 0 \)).
- If \( c \) is a scalar, then \( c \cdot u = \begin{pmatrix} cx \\ cy \end{pmatrix} \).

#### Consider the cases for \( c \):
1. **If \( c > 0 \)**: 
   - The signs of \( x \) and \( y \) remain the same, so \( (cx)(cy) = c^2(xy) \). Since \( c^2 > 0 \), \( (cx)(cy) \) will still be non-negative. Hence, \( c \cdot u \) is in \( W \).

2. **If \( c = 0 \)**:
   - \( c \cdot u = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \), which is the zero vector. The zero vector is trivially in \( W \) because \( 0 \times 0 = 0 \geq 0 \).

3. **If \( c < 0 \)**:
   - The signs of \( cx \) and \( cy \) are opposite to those of \( x \) and \( y \), but \( (cx)(cy) = c^2(xy) \). Since \( c^2 > 0 \), the product is still non-negative. Thus, \( c \cdot u \) is still in \( W \).

### Conclusion:
Regardless of whether \( c \) is positive, negative, or zero, the product \( (cx)(cy) = c^2(xy) \) remains non-negative, which means \( c \cdot u \) is still in \( W \).

**Correct Answer: Option A** is correct because it correctly explains that \( c \cdot u \) will always be in \( W \) because \( c^2(xy) \geq 0 \) whenever \( xy \geq 0 \).

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### Do you want more details or have any questions?

### Related Questions:
1. What would change if \( W \) were defined differently, such as only including one quadrant?
2. How does the union of the first and third quadrants affect the properties of \( W \)?
3. What does it mean for a set to be closed under scalar multiplication?
4. Can you provide an example of a set that is not closed under scalar multiplication?
5. How would the answer change if the condition \( xy > 0 \) were used instead?

### Tip:
When dealing with vector spaces or sets defined by inequalities, always check the conditions for each quadrant and the effect of scalar multiplication on those conditions.

Explore vector operations in quadrants of the xy-plane with a focus on scalar multiplication. Analyze why scalar multiples of vectors in specific quadrants remain within those regions. Suitable for advanced high school students and anyone interested in linear algebra concepts.

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