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.
