Pre-calculus and vectors; please help?

Prove that if vector u is orthogonal to vectors v and w, then u is orthogonal to cv+dw


where c and d are anyt scalars.





I do not even understand the question. How can u be orthogonal to both w and v? in the diagram, w and u are clearly orthogonaly, but u and v form an obtuse angle with w between them....

Pre-calculus and vectors; please help?
Don't be fooled by the pictures...they may be in 3-D, but it is hard to draw that in a book. It does work, though:





u = %26lt;u1, u2%26gt;


v = %26lt;v1, v2%26gt;


w = %26lt;w1, w2%26gt;





Multiplying the appropriate vectors by the appropriate constants:


cv = %26lt;cv1, cv2%26gt;


dw = %26lt;dw1, dw2%26gt;





Add them together:


cv + dw = %26lt;cv1+dw1, cv2+ dw2%26gt;





Definition of dot product:


x dot y = x1y1 + x2y2





Use that on u dot (cv + dw):


u dot (cv + dw) = (u1)(cv1 + dw1) + (u2)(cv2 + dw2)





Distribute:


u1cv1 + u1dw1 + u2cv2 + u2dw2





Pull the constants out in front:


c(u1v1) + d(u1w1) + c(u2v2) + d(u2w2)





Rearrange things so all the constant terms are hanging out together:


c(u1v1) + c(u2v2) + d(u1w1) + d(u2w2)





Factor the constants out:


c(u1v1 + u2v2) + d(u1w1 + u2w2)





In the given statements,


u dot v = u1v1 + u2v2 = 0


u dot w = u1w1 + u2w2 = 0





Substitute and solve:


c(0) + d(0)


0 + 0


0





Since the dot product equals zero, the two vectors are orthogonal