Hi , I have a question stated as Given the vector B=-6x-8y+9z and vector C= 5x-3y+4z . Find vectors T1 and T2 such that T1 is parallel to vector C and perpendicular to vector T2. where vector B = T1 + T2 . So far, i was able to find a vector T1 which is parallel to vector C but couldnt figure out how i can make it perpendicular to the vector T2 because when i try to make it perpendicular to the vector T2, it becoms impossible to satisfy the given equation B = T1 + T2 .
Since T1 is parallel to vector so i made it to be 2C. T1=10x-6y+4z. Now according to the given eq. B = T1 + T2 if i put values of T1 and B here , i get the vector T2 but the problem is that the resulting T2 vector when multiplied witht T1 vector as a dot product doesnot gives a zero. T1 and T2 multiplication should result in 0 since they are supposed to be perpendicular.
Your problem is in assuming that you can arbitrarily pick ANY vector for T1 that happens to be parallel to C. What makes you think you can assume that? What if I were to ask you to find a number A such that A is negative and A multiplied by itself is +25? Would you pick A to be -10 since that is negative and then conclude that the problem has no solution since (-10)(-10) = 100 and not 25? It's the same thing here. Requiring T1 to be parallel to C places a constraint on the direction, not the magnitude. But the magnitude isn't arbitrary -- it must be chosen so as to satisfy the other contraints. Sketch out the problem and the solution in 2D. Just pick two vectors for B and C that are at some acute angle to each other. You'll quickly see what the relationships between T1 and T2 are and how they relate to B and C.