If the resultant of the vector 3i+4j+5k and 5i +3j+ 4k makes an angle theta with x axis the find cos theta.?

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Solution

Let A=3i+4j+5k and B=5i+3j+4k

then the resultant C, can be written as

C=A+B=8i+7j+9k

As we know a unit vector along x-axis is i,

Hence the required angle # between C and i can be given as

Cosθ = {C.i}/C.i [C and i are magnitudes of C and i , This relation comes from the dot product of vectors where A.B = AB.Cosθ → Cosθ =A.B/AB]

Now we need to know the components on RHS.

C = √(8²+7²+9²) =√(64+49+81) = √194 = 13.93

i = 1

C.i = (8i+7j+9k).i = 8+0+0 = 8

So now Cos θ= {C.i}/C.i = 8/13.93 = 0.574

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