Need help with 2 multivariable calculus problems please?

Question

A) The question is [let u and v be vectors]:
prove that ||u+v||^2 + ||u-v||^2=2||u||^2+2||v||^2
and interpret the result geometrically by translating it into a theorem about parallelograms.

I went about proving the statement component wise, resulting in 2(u1)^2+2(u2)^2+ 2(v1)^2 + 2(v2)^2 = 2(u1)^2+2(u2)^2+ 2(v1)^2 + 2(v2)^2; however I don't know how to translate that into a theorem about parallelograms. Am I even attacking this problem in the correct manner?

B) Find the parametric equations for the line that is perpendicular to L1: x = 1 + 2t, y= 2-t, z=4-2t and L2: x= 9+t, y=5+3t, z = -4-t, and passes through their point of intersection

I found the point of intersection at (7, -1, -2) and I know that there is no y direction, but I cannot seem to find the x and z direction. Pretty much I need to find a and b in the following line but i do not know how.
x= 7 + At
y= -1
z= -2 + Bt

Any help on either would really be appreciated! Thanks!

Answer 1

A) Draw a diagram showing u and v and draw u+v and u-v. ||u+v||^2 + ||u-v||^2 is the squre of the hypotenuse joining u+v and u-v, yes they are at right angles.

B) convert L1 and L2 into vector form.

L1 = [1+2t,2-t,4-2t]
L2 = [9+t,5+3t,-4-t]

Now find the cross product of L1 and L2.

Answer 2

possibly if I do a million of them you will see a thank you to do the different ones. With a multivariable function, you gotta see if the shrink is a similar whilst the component techniques the shrink component alongside distinctive paths. one thank you to describe distinctive paths is to look at strains: y = mx the place m is a few consistent - then additionally look on the trails like x=0. My textbook would not incorporate a data that observing all linear paths is adequate. it style of feels to me like it is adequate nevertheless. lim x-> 0 : (x^2)(m^2x^2) / (x^4 + m^4 x^4) = lim x->0 m^2 x^4 / (m^4+a million) x^4 = m^2 / (m^4 + a million) i might say the opportunities for m^2 / (m^4 + a million) being consistent whilst m transformations to distinctive values are somewhat slender, so for this one the shrink would not exist. For the final one i think of you ought to use distinctive strains in direction of the muse in 3 dimensions like (x=x, y=mx, z=px) and then the shrink will come out in terms of m and p.