find a vector with perpendicular

Question

Find the real number a such that the vector w = au+v is perpendicular to 2u-v

Im trying to solve this but I dont know how to do this ... could anyone please help me?

u = 2i + 2j + k and v = 3i - j - 2k.

Answer 1

I will show you how to solve this, and you can fill in the math.

 

If w is perpendicular to 2u - v, then the dot product of w and 2u - v is zero.

 

w = au + v = a(2i + 2j + k) + (3i - j - 2k) = (2a + 3)i + (2a - 1)j + (a - 2)k

 

Similarly, expand 2u - v in terms of i, j, and k components.  Then evaluate the dot product.

 

w·(2u - v) = 0

 

Recall the dot product of 2 vectors A and B is

 

A·B = A_xB_x + A_yB_y + A_zB_z

 

This will give an equation whose only unknown is a, and you can then solve for a.  Can you finish from here?