The book states
Three Vectors
U= Uxi + 3j+2k
V= -3i + Vyj+3k
W= -2i + 4j+Wzk
are mutally perpendicular Use dot product to solve for the componets
Ux,Vy & Wz
I do not know how to begin and am pretty confused the answer are
Ux=2.875
Vj= .857
Wz=-3.143
Can anyone help??
You have three unknowns Ux, Vy and Wz. You are told that they areOriginally Posted by jaycas21
mutually perpendicular, and you are to use the dot product.
The dot product of perpendicular vectors is zero, so you have:
U.V=0,
U.W=0,
V.W=0.
Form these dot products and you will have three linear simultaneous
equations which you then need to solve.
Dot product of two vectors:$\displaystyle x_1i+x_2j+x_3k$ and $\displaystyle y_1i+y_2j+y_3k$ is:
$\displaystyle x_1y_1+x_2y_2+x_3y_3$,
where $\displaystyle i$, $\displaystyle j$ and $\displaystyle k$ are the unit vectors along a set of perpendicular axes.
RonL
If two vectors are perpendicular, then their dot product is zero.
Since the 3 given vectors are mutually perpendicular, then,
U dot V = 0
U dot W = 0
V dot W = 0
U dot V = (Ux)(-3) +3(Vy) +2(3) = 0
-3Ux +3Vy +6 = 0
-Ux +Vy +2 = 0 --------------(1)
U dot W = (Ux)(-2) +3(4) +2(Wz) = 0
-2Ux +12 +2Wz = 0
-Ux +6 +Wz = 0 -------------(2)
V dot W = -3(-2) +(Vy)(4) +3(Wz) = 0
6 +4Vy +3Wz = 0 --------(3)
3 equations, 3 unknowns, solvable.
(2) minus (1),
6 +Wz -Vy -2 = 0
Wz = Vy -6 +2
Wz = Vy -4 --------------(i)
Substitute that into (3),
6 +4Vy +3(Vy -4) = 0
6 +4Vy +3Vy -12 = 0
7Vy -6 = 0
7Vy = 6
Vy = 6/7 = 0.857 ----------answer.
Substitute that into (i),
Wz = 0.857 -4 = -3.143 ---------answer.
Substitute the 0.857 for Vy into (1),
-Ux +Vy +2 = 0 -----(1)
-Ux +0.857 +2 = 0
-Ux +2.857 = 0
Ux = 2.857 --------------answer.
------------------------
Zeez, late again.
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