Components of Vectors Question Help?

Question

A certain vector in the xy plane has a y component = 10 & x component = 4. The vector is rotated in the plane so that the x component is doubled. What is the new y component?

I don't get the concept of this question, can anyone provide an explanation to me? thanks

Answer 1

y = R sin(theta) and x = R cos(theta); where R is a vector at angle theta wrt the X axis and magnitude abs(R).

y and x form a right triangle; so that R^2 = y^2 + x^2 is true. Now set X = 2x, which means the new x, X, is twice the old x = 4; so that R^2 = Y^2 + X^2 = y^2 + x^2 = Y^2 + 4x^2. Then Y^2 = R^2 - 4x^2 = y^2 + x^2 - 4x^2 = 100 + 16 - 4*16 = 100 - 3*16 = 52; so that Y = sqrt(52) which is a bit over 7, which is the new y component Y when X = 2x = 8.

The important lesson here is that R = constant no matter what the orientation of theta. Thus R^2 = constant = Y^2 + X^2 or = y^2 + x^2, which means an increase in x or X has to be offset by a decrease in y or Y.

Answer 2

v_hat = v/|v| = / ?(a million + 4 + 4) = |u| = ?(6² + 3² + 2²) = ?(36 + 9 + 4) = 7 comp_v(u) = u • v_hat = <6,3,2> • = (2 - 2 - 4/3) = -4/3