Given that OD=kOA find the value of k for which OD+OB is parallel to the x axis

Parralel vectors
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Let OA=a1i+a2j, then OD=ka1i+ka2j, and let OB=b1i+b2j. (i and j are the unit vectors in the orthogonal x and y directions respectively. a1, a2, b1, b2 are scalar quantities.)

OD+OB=(ka1+b1)i+(ka2+b2)j. If this resultant vector is parallel to the x-axis then the y-component must be zero, so ka2+b2=0, k=-b2/a2. b2=OB.j and a2=OA.j, so k=-OB.j/OA.j. We have no more information about the vectors.

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