Question

Find vector perpendicular to another vector a+b and a-b

Answer 1

let the known vector be P=ai+bj+ck.........................(1)

and, let the unknown vector be Q=xi+yj+zk..................(2)

Since the two vectors are to be perpendicular to each other,their dot product should be 0.

ie : P.Q=0=(ai+bj+ck).(xi+yj+zk)=ax+by+cz=0.........(3)

Now we have three variables and one equation. So there exists infinitely many solutions.

To find one of them, assign any value to any two variables of x,y and z.

This will give you the third variable when you solve the above equation.

Then you get a vector when you plugin the values of x,y and z to the Q

equation (2).

then you have found a vector which satisfies the condition given in the question.

You may find vectors of any magnitude that still satisfies the condition by multiplying a suitable scalar to the newly found vector Q.

Note that there are infinitely many solutions if there is only these two conditions. To find a unique vector, you must have at least three independent equations.