The sum of two vectors is the largest when the two vectors are

The sum of two vectors is largest when both vectors are parallel and share the same direction.

When adding two vectors, the result will always be a new vector which depends on 3 things, the modulus of the two initial vectors and the angle between them . Given two random vectors, their sum will be largest when the modulus of the result will be maximum.

Suppose we start with 2 random vectors of modulus A and B. The modulus of their sum, let's call it C, will be expressed by the cosine theorem.

Where is the angle between the 2 initial vectors. From the above equation we can see that C will be maximum when the cosine of will be minimum, this happens when both vectors are parallel and share the same direction ( would be zero and ).

As further knowledge, C will be minimum when the cosine of will be maximum, this happens when both vectors are parallel but have opposite direction ( would be equal to and ).

Cosine theorem, addition of vectors, maximum