Let u , v , w be the vectors such that | u |=1,| v |=2,| w |=3 . If the projection v along u is equal to that of w along u and the vectors v , w are perpendicular to each other then | u - v + w | equals

Step-by-step solution

Given vectors u , v , w with magnitudes | u |=1 , | v |=2 , | w |=3 , and conditions that the projection of v and w along u are equal, and that v and w are perpendicular. The projection equality implies v u | u | = w u | u | , so v u = w u . Since v w = 0 , we compute the squared magnitude: | u - v + w |^2 = ( u - v + w ) ( u - v + w ) = | u |^2 + | v |^2 + | w |^2 - 2( u v ) + 2( u w ) - 2( v w ) = 1 + 4 + 9 - 2( u v ) + 2( u v ) - 0 = 14 Thus | u - v + w | = 14 , matching option A.