Find the value of g such that vector x=(-5,30,g) is perpendicular to both vector m=(8,1,2) and vector n=(3,0,5).

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The vector x = <-5, 30, g> is perpendicular to m = <8 , 1, 2> and n = <3, 0, 5>

The cross product of m and n is equal to x.

The cross product of A = A1*i + A2*j + A3*k and B = B1*i + B2*j + B3*k is given by the vector < A2*B3 - A3*B2, A3*B1 - A1*B3, A1*B2 - A2*B1>

Here A2*B3 - A3*B2 = 1*5 - 2*0 = 5

A3*B1 - A1*B3 = 2*3 - 8*5 = 6 - 40 = -34

A1*B2 - A2*B1 = 8*0 - 1*3 = -3

This gives x =< 5, -34, -3>

As the direction of x can be reversed also x = <-5, 34, 3>

It is not possible to have a vector of the form <-5, 30, g> perpendicular to m and n.

The vector perpendicular to the two is either <-5, 34, 3> or < 5, -34, -3>

As the x-component is -5, the value of g is 3 (but the y-component is 34 not 30).

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