if vector a is fixed but vector b can be rotated in any direction. what should be the angle of vector b to give the maximum resultant sum
same direction
To find the angle of vector b that gives the maximum resultant sum, we can use the concept of vector addition and trigonometry.
The resultant sum of two vectors is maximized when they are aligned in the same direction. In this case, vector a is fixed, so we want to align vector b with vector a to maximize the resultant sum.
Let's assume that the angle between vector a and vector b is θ. To maximize the resultant sum, we need to find the value of θ that makes the two vectors aligned.
We can use the dot product of vector a and vector b to determine their alignment. The dot product of two vectors a and b is given by:
a · b = |a| |b| cos(θ)
The dot product gives us the cosine of the angle between the two vectors. Since we want them to be aligned, the angle should be 0°, which means cos(θ) = 1.
Therefore, we have:
|a| |b| cos(θ) = |a| |b|
Canceling out |a| and |b|, we get:
cos(θ) = 1
To find the angle θ that satisfies this equation, we take the inverse cosine (or arccosine) of both sides:
θ = arccos(1)
The arccosine of 1 is 0. Therefore, the angle of vector b that gives the maximum resultant sum is 0°, where vector b is aligned with vector a.