Sure, let's analyze each of the given conditions step-by-step:
### Condition A: [tex]\( v \cdot w = 40 \)[/tex]
The dot product of two vectors [tex]\( v \)[/tex] and [tex]\( w \)[/tex] is given by:
[tex]\[
v \cdot w = v_1 \cdot w_1 + v_2 \cdot w_2
\][/tex]
For [tex]\( v = (8, -4) \)[/tex] and [tex]\( w = (-4, 2) \)[/tex]:
[tex]\[
v \cdot w = (8) \cdot (-4) + (-4) \cdot (2) = -32 + (-8) = -40
\][/tex]
So, [tex]\( v \cdot w \neq 40 \)[/tex], but instead [tex]\( v \cdot w = -40 \)[/tex]. Thus, Condition A is false.
### Condition B: The [tex]\( y \)[/tex]-component of [tex]\( w \)[/tex] is [tex]\( 2 e_2 \)[/tex]
For [tex]\( w = (-4, 2) \)[/tex], the [tex]\( y \)[/tex]-component of [tex]\( w \)[/tex] is 2.
Hence, Condition B is true.
### Condition C: The [tex]\( x \)[/tex]-component of [tex]\( v \)[/tex] is [tex]\( 4 e_1 \)[/tex]
For [tex]\( v = (8, -4) \)[/tex], the [tex]\( x \)[/tex]-component of [tex]\( v \)[/tex] is 8.
Since the given condition states that the [tex]\( x \)[/tex]-component of [tex]\( v \)[/tex] should be [tex]\( 4 e_1 \)[/tex], where [tex]\( e_1 \)[/tex] presumably represents the unit vector in the x-direction, and [tex]\( 8 \)[/tex] is not equal to [tex]\( 4 \)[/tex], Condition C is false.
### Condition D: [tex]\( v = -2 w \)[/tex]
To check if [tex]\( v = -2 w \)[/tex], we need to scale [tex]\( w \)[/tex] by -2 and see if we get [tex]\( v \)[/tex].
[tex]\[
-2 w = -2 (-4, 2) = (8, -4)
\][/tex]
This is indeed equal to [tex]\( v \)[/tex]. Hence, Condition D is true.
### Summary
- Condition A is false: [tex]\( v \cdot w \neq 40 \)[/tex].
- Condition B is true: The [tex]\( y \)[/tex]-component of [tex]\( w \)[/tex] is indeed 2.
- Condition C is false: The [tex]\( x \)[/tex]-component of [tex]\( v \)[/tex] is 8, not 4.
- Condition D is true: [tex]\( v \)[/tex] is indeed [tex]\( -2 w \)[/tex].
So, the correct statements are B and D.
