How does multiplying a vector by a scalar value of 2pi change the vector?
A.) The vector will change direction and increase in magnitude.
B.) The vector will change direction and decrease in magnitude.
C.) The vector will not change direction but will increase in magnitude.
D.) The vector will not change direction but will decrease in magnitude.
I think its A
The correct answer is C.) The vector will not change direction but will increase in magnitude.
When you multiply a vector by a scalar value of 2π, you are essentially scaling the vector by that value. This means that the magnitude of the vector will become 2π times larger, while the direction of the vector remains unchanged. So, the vector will increase in magnitude but will not change its direction.
To understand how multiplying a vector by a scalar value of 2π changes the vector, we need to consider the effect of scalar multiplication on both the magnitude and direction of the vector.
Scalar multiplication involves multiplying each component of the vector by the scalar value. In this case, multiplying the vector by 2π means multiplying each component of the vector by 2π.
If the original vector has a magnitude of m and a direction of θ, then multiplying it by 2π results in a new vector with a magnitude of (2π) * m and the same direction of θ.
Now, let's analyze the options:
A.) The vector will change direction and increase in magnitude.
This option implies that multiplying by 2π changes the direction of the vector. However, we established that only the magnitude changes, not the direction. So, this option is incorrect.
B.) The vector will change direction and decrease in magnitude.
This option also implies a change in direction, which we know is not the case. Additionally, scalar multiplication with a positive value like 2π will result in an increase in magnitude, not a decrease. So, this option is also incorrect.
C.) The vector will not change direction but will increase in magnitude.
This option correctly states that the direction of the vector remains unchanged. Additionally, multiplying the magnitude of the vector by 2π will indeed result in an increase in magnitude. So, this option is the correct answer.
D.) The vector will not change direction but will decrease in magnitude.
This option is incorrect because it claims a decrease in magnitude, whereas multiplying by 2π actually increases the magnitude.
Therefore, the correct answer is C.) The vector will not change direction but will increase in magnitude.
Why not try it with an actual example to see what happens?
e.g.
u = <3,4> find its magnitude and direction
v = 6u = <18,24> find its magnitude and direction
What happened?