In the cuboid abcd-a'b'c'd ', P and R are the moving points on BB' and CC ', respectively. When P and R satisfy what conditions, PR is parallel to the plane ab'd

In the cuboid abcd-a'b'c'd ', P and R are the moving points on BB' and CC ', respectively. When P and R satisfy what conditions, PR is parallel to the plane ab'd

When PR ∥ ad, PR ∥ surface ab'd,
So: when BP / PB '= Cr / RC', PR ‖ plane ab'd

Given that the angle between the diagonal a'c of the cuboid abcd-a'b'c'd 'and the side edge BB' is 45 degrees, and ab = BC = 1, find the angle between a'c and the side b'c'cc

Surface ABCD is square, surface aa1c1c is square
Connect BC1, ∵ ab ⊥ bb1c1c ∪ ac1b as the angle
Let AB = 1, then AC1 = 2, BC1 = √ 3, so ∠ ac1b = 30 degree

In a cuboid, find the distance from BC to plane AB ` c ` d (a ` B ` c ` d below, ABCD below)
AA 'is a, AB is B

Because BC / / AD, ad belongs to plane AB ` c ` D,
So BC / / plane AB ` c ` d
Through point B, make BF vertical ab 'and cross ab' to point F,
B ` c ` vertical plane ABB ` a ', so B ` c ` vertical BF
So BF vertical plane AB ` c ` d
That is, the distance from BC to plane AB ` c ` d d d = BF = AB * BB '/ ab' = b * A / [radical (a ^ 2 + B ^ 2)]

Given that the points E and F of the rectangle abcd-a'b'c'd 'are the centers of the upper and lower surface a'b'c'd' and the plane cc'd'd respectively, find the value of XYZ in the following questions
（1）AC'=xAB+yBC+zCC'
(2)AE=xAB+yBC+zCC'
(3)AF=xBA+yBC+zCC'
The capital letters are vectors

（1）AC'=xAB+yBC+zCC'AC'=AB+BC+CC'∴x=y=z=1(2)AE=xAB+yBC+zCC'AE=AA'+A'E=CC'+1/2(A'C')=CC'+1/2AC=1/2AB+1/2BC+CC'x=y=1/2,z=1(3)AF=xBA+yBC+zCC'AF=AD...