The sum of two vectors of fixed magnitudes has the greatest magnitude when the angle between these two vectors are?
zero
The sum of two vectors of fixed magnitudes has the greatest magnitude when they are collinear or parallel to each other. In other words, the angle between the two vectors is 0 degrees (or 180 degrees) when the magnitude of their sum is maximized.
To find the angle between two vectors that yields the greatest magnitude when their sum is taken, we can use the concept of vector addition and the properties of scalar and vector products.
Let's assume we have two vectors, A and B, with fixed magnitudes. The sum of these two vectors is C, given by the equation:
C = A + B
The magnitude of a vector can be found using the Pythagorean theorem, which states that the square of the magnitude is equal to the sum of the squares of its components:
|C|^2 = C · C
Using the properties of the dot product, we can rewrite the equation as:
|C|^2 = (A + B) · (A + B)
Expanding the dot product using the distributive property, we get:
|C|^2 = A · A + A · B + B · A + B · B
Since the dot product is commutative, we can write:
|C|^2 = A · A + 2(A · B) + B · B
Now, since we want to find the angle that maximizes the magnitude of C, we need to maximize |C|^2. This means we need to maximize the term 2(A · B).
To maximize a dot product, we can use the fact that the dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them:
A · B = |A| |B| cos(theta)
Therefore:
2(A · B) = 2 |A| |B| cos(theta)
To maximize this expression, we want the angle theta to be cos(theta) = 1, which occurs when theta = 0 degrees.
So, the angle between the two vectors that yields the greatest magnitude when their sum is taken is 0 degrees, or when the two vectors are perfectly aligned.