A vector makes an angle of 45 degree with x and y axes. What is the angle formed by the vector with z axis?

Answer is 90 degree. See how?

you know that direction that direction cosines of a vector, say r are nothing but cosines of the anges made by the vector with the three coordinate axes i.e. X-axis , Y-axis, Z-axis with basic  the property that cos² A + cos² B + cos²C=1

where A, B, C are the respective angles made by the vector with the axes

You say that the vector makes anles 45 degree with X and Y axis therefore cos²45 + cos² 45 +cos²C=1

This implies that (1/2)²  + (1/2)²² +cos²C=1 or ½  + ½  + cos²C=1 => cos²C=0 => cos C=0 => C=/2

so anglw made by the vector is /2 with Z-axis

 

kkbisht