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

Step-by-step solution

Given | u | = 1 , | v | = 2 , and | w | = 3 . The projection of v along u is equal to the projection of w along u . v u | u | = w u | u | v u = w u Since v and w are perpendicular to each other, their dot product is zero: v w = 0 We need to find the value of | u - v + w | . Consider its square: | u - v + w |^2 = | u |^2 + |- v |^2 + | w |^2 + 2( u (- v )) + 2((- v ) w ) + 2( u w ) | u - v + w |^2 = | u |^2 + | v |^2 + | w |^2 - 2( u v ) - 2( v w ) + 2( u w ) Substituting u v = u w and v w = 0 into the expansion: |