Suppose vector 1, vector 2, vector 3 is a set of vectors mutually perpendicular. Assume that Length of vector v1 = sqrt(35) length of vector 2 = sqrt(94) length vector v3 = sqrt(36) Let w be a vector in Span(v1,v2,v3) such that (vector w)dot product(vector v1) = 35 (vector w)dot product(vector v2) = -188 (vector w)dot product(vector v3) = 36

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As   W is a vector in span (V_1, V_2, V_3, V_4).

Also given mod(V_1)=sqrt(35)

                mod(V_2)=sqrt(94)

                mod(V_3)=sqrt(36)

Now w can be written as w=projection of w on V_1+projection of w on V_2+projection of w on V_3

So, projection of w on V_1={(w.V_1)/(mod(V_1))^2}.V_1

or,  projection of w on V_1=(35/35)V_1=1V_1

Simlarly...

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As   W is a vector in span (V_1, V_2, V_3, V_4).

Also given mod(V_1)=sqrt(35)

                mod(V_2)=sqrt(94)

                mod(V_3)=sqrt(36)

Now w can be written as w=projection of w on V_1+projection of w on V_2+projection of w on V_3

So, projection of w on V_1={(w.V_1)/(mod(V_1))^2}.V_1

or,  projection of w on V_1=(35/35)V_1=1V_1

Simlarly projection of w on V_2=(-188/94)V_2=-2V_2

and     projection of w on V_3=(36/36)V_3=1V_3

so, w=V_1-2V_2+V_3

Answer.

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