To determine the relationship between the direction of vector [tex]\( b \)[/tex] and vector [tex]\( 2b \)[/tex] when the scalar is 2, let's go through the steps step-by-step.
1. Identify the given vector [tex]\( b \)[/tex]:
[tex]$ b = \langle -3, 5 \rangle $[/tex]
2. Perform the scalar multiplication:
When you multiply the vector [tex]\( b \)[/tex] by the scalar 2, each component of the vector is multiplied by 2:
[tex]$ 2b = 2 \times \langle -3, 5 \rangle $[/tex]
This results in:
[tex]$ 2b = \langle 2 \times (-3), 2 \times 5 \rangle = \langle -6, 10 \rangle $[/tex]
3. Compare the original vector [tex]\( b \)[/tex] and the scaled vector [tex]\( 2b \)[/tex]:
- The original vector [tex]\( b \)[/tex] is [tex]\( \langle -3, 5 \rangle \)[/tex].
- The new vector [tex]\( 2b \)[/tex] after scalar multiplication is [tex]\( \langle -6, 10 \rangle \)[/tex].
4. Examine the direction:
- Both vectors [tex]\( \langle -3, 5 \rangle \)[/tex] and [tex]\( \langle -6, 10 \rangle \)[/tex] have the same directional ratio. This means if you draw them on a coordinate plane, they point in the same direction but the magnitudes differ.
- Scalar multiplication by a positive number (2 in this case) does not change the direction of the vector but just scales its magnitude.
5. Conclusion:
The direction of the vector [tex]\( b \)[/tex] and the vector [tex]\( 2b \)[/tex] remain the same. Therefore, the correct option from the listed choices is:
- they are the same
So, the relationship between the direction of vector [tex]\( b \)[/tex] and vector [tex]\( 2b \)[/tex] is that they are the same.
