The sum of the scalar products u.v + v.w + w.u given that u + v + w = 0 and the magnitudes of the vectors u, v, w are 3, 4, and 5 respectively, is 50.
The question provided relates to vector addition and the scalar (dot) product between vectors. Given that u, v, and w are vectors such that u+v+w=0, we know that these vectors form a triangle that closes back on itself. Because of this relationship, we can infer that they cancel each other out when added together, resulting in a vector sum of zero.
The scalar product of two vectors is defined by the formula u.v = |u|*|v|*cos(theta), where theta is the angle between the vectors u and v. The question asks for the sum of the scalar products u.v + v.w + w.u. This can be simplified using the vector addition equation provided and the properties of scalar products.
By the given vector magnitudes, we have |u|=3, |v|=4, and |w|=5. Using the vector addition equation, we can express w as -u-v and substitute this into the scalar products. Given the symmetry of the dot product, the sum of the scalar products is equivalent to the negative of the sum of the squares of the magnitudes of the vectors, which can be calculated as - ( |u|² + |v|² + |w|² ), resulting in -(-|-|).
In conclusion, the sum u.v + v.w + w.u is equal to - ( |u|² + |v|² + |w|²). Plugging in the given magnitudes, we get - (-3² - 4² - 5²), simplifying to - ( -50 ), which is 50.