Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
Let's carefully examine each of the given statements to determine their validity:
Statement A: [tex]\(w = -2v\)[/tex]
To check this, we can verify if [tex]\( w \)[/tex] is equal to [tex]\(-2\)[/tex] times [tex]\( v \)[/tex]. The given vectors are:
[tex]\[ v = (5, -2) \][/tex]
[tex]\[ w = (-10, 4) \][/tex]
Now, let's calculate [tex]\(-2 \times v\)[/tex]:
[tex]\[ -2 \times v = -2 \times (5, -2) = (-10, 4) \][/tex]
Since [tex]\( w = (-10, 4) \)[/tex] matches [tex]\(-2 \times v \)[/tex], this statement is True.
Statement B: The [tex]\( y \)[/tex]-component of [tex]\( w \)[/tex] is [tex]\( 4 e_2 \)[/tex]
The [tex]\( y \)[/tex]-component of [tex]\( w \)[/tex] is 4. In a two-dimensional space, [tex]\( e_2 \)[/tex] represents the unit vector along the y-axis, which is [tex]\( (0, 1) \)[/tex]. Thus, [tex]\( 4 e_2 \)[/tex] means 4 times the unit vector along the y-axis:
[tex]\[ 4 \times (0, 1) = (0, 4) \][/tex]
Here, the [tex]\( y \)[/tex]-component of [tex]\( w = (-10, 4) \)[/tex] is indeed 4, so this statement is True.
Statement C: [tex]\( v \cdot w = -58 \)[/tex]
The dot product of [tex]\( v \)[/tex] and [tex]\( w \)[/tex] is computed as follows:
[tex]\[ v \cdot w = (5 \times -10) + (-2 \times 4) \][/tex]
[tex]\[ v \cdot w = -50 - 8 \][/tex]
[tex]\[ v \cdot w = -58 \][/tex]
Since the calculation confirms this value, this statement is True.
Statement D: [tex]\( v \)[/tex] and [tex]\( w \)[/tex] are perpendicular
Two vectors are perpendicular if their dot product is zero. We already computed the dot product:
[tex]\[ v \cdot w = -58 \][/tex]
Since [tex]\(-58 \neq 0\)[/tex], the vectors [tex]\( v \)[/tex] and [tex]\( w \)[/tex] are not perpendicular. Thus, this statement is False.
In summary, the statements which are true are:
- A. [tex]\( w = -2 v \)[/tex]
- B. The [tex]\( y \)[/tex]-component of [tex]\( w \)[/tex] is [tex]\( 4 e_2 \)[/tex]
- C. [tex]\( v \cdot w = -58 \)[/tex]
And the false statement is:
- D. [tex]\( v \)[/tex] and [tex]\( w \)[/tex] are perpendicular.
Statement A: [tex]\(w = -2v\)[/tex]
To check this, we can verify if [tex]\( w \)[/tex] is equal to [tex]\(-2\)[/tex] times [tex]\( v \)[/tex]. The given vectors are:
[tex]\[ v = (5, -2) \][/tex]
[tex]\[ w = (-10, 4) \][/tex]
Now, let's calculate [tex]\(-2 \times v\)[/tex]:
[tex]\[ -2 \times v = -2 \times (5, -2) = (-10, 4) \][/tex]
Since [tex]\( w = (-10, 4) \)[/tex] matches [tex]\(-2 \times v \)[/tex], this statement is True.
Statement B: The [tex]\( y \)[/tex]-component of [tex]\( w \)[/tex] is [tex]\( 4 e_2 \)[/tex]
The [tex]\( y \)[/tex]-component of [tex]\( w \)[/tex] is 4. In a two-dimensional space, [tex]\( e_2 \)[/tex] represents the unit vector along the y-axis, which is [tex]\( (0, 1) \)[/tex]. Thus, [tex]\( 4 e_2 \)[/tex] means 4 times the unit vector along the y-axis:
[tex]\[ 4 \times (0, 1) = (0, 4) \][/tex]
Here, the [tex]\( y \)[/tex]-component of [tex]\( w = (-10, 4) \)[/tex] is indeed 4, so this statement is True.
Statement C: [tex]\( v \cdot w = -58 \)[/tex]
The dot product of [tex]\( v \)[/tex] and [tex]\( w \)[/tex] is computed as follows:
[tex]\[ v \cdot w = (5 \times -10) + (-2 \times 4) \][/tex]
[tex]\[ v \cdot w = -50 - 8 \][/tex]
[tex]\[ v \cdot w = -58 \][/tex]
Since the calculation confirms this value, this statement is True.
Statement D: [tex]\( v \)[/tex] and [tex]\( w \)[/tex] are perpendicular
Two vectors are perpendicular if their dot product is zero. We already computed the dot product:
[tex]\[ v \cdot w = -58 \][/tex]
Since [tex]\(-58 \neq 0\)[/tex], the vectors [tex]\( v \)[/tex] and [tex]\( w \)[/tex] are not perpendicular. Thus, this statement is False.
In summary, the statements which are true are:
- A. [tex]\( w = -2 v \)[/tex]
- B. The [tex]\( y \)[/tex]-component of [tex]\( w \)[/tex] is [tex]\( 4 e_2 \)[/tex]
- C. [tex]\( v \cdot w = -58 \)[/tex]
And the false statement is:
- D. [tex]\( v \)[/tex] and [tex]\( w \)[/tex] are perpendicular.
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.