The three vectors: U = U x i + 3j + 2k V = -3i + V y j + 3k W = -2i +4j + W z k

Question

The three vectors:

U = Uxi + 3j + 2k

V = -3i + Vyj + 3k

W = -2i +4j + Wzk

are mutually perpendicular (meaning that the dot product of two vectors equals 0).

Use the dot product to determine the components Ux, Vy, Wz.

I started out by trying to find the dot product of U and V and then solving it like a system of equations but everything cancels out. I am not sure what else to do, or if I am simply solving the system wrong.

Also, if this helps, the answers in the back of the book state that Ux = 2.857, Vy = 0.857, and Wz = -3.143

Explanation / Answer

U.V = 0 or ( Uxi + 3j + 2k ) .( -3i + Vyj + 3k ) = 0 or -3 Ux + 3 Vy + 6 = 0 ----------------(1) U.W = 0 or ( Uxi + 3j + 2k). (-2i +4j + Wzk) = 0 or -2 Ux + 12 + 2 Wz = 0 ----------------(2) V.W = 0 or (-3i + Vyj + 3k).(-2i +4j + Wzk)=0 or 6 + 4 Vy + 3 Wz = 0 ----------------(3) solving 1 2 and 3 -3 Ux + 3 Vy + 6 = 0 ----------------(1) -2 Ux + 12 + 2 Wz = 0 ----------------(2) 6 + 4 Vy + 3 Wz = 0 ----------------(3) 3 variables , 3 unknowns , so this is solvable. Ux = 0 , Vy=0 , Wz = 0 ; hence the answer is the components are zero, [ the components can be zero ]

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