Problems: Dot product of vectors

[bubbles]

Q1.

A parallelepiped is an oblique prism that has a parallelogram cross-section It has three pairs of parallel and congruent faces. OABCDEFG is a parallelepiped with
-   OA = 3j
-   OC = -i + j +2k
-   OD = 2i – j
-   Show that the diagonals DB and CE bisect each other, and find the acute angle between them

Where have I gone wrong?
1.   Find DB
-   DB = DG + GC + CB
-   *DG=OC and GC=-OD and CB=OA
-   DB = (-i + j +2k) – (2i – j) + 3j
            = -i + j + 2k - 2i + j + 3j
            = -i – 2i + j + 3j + j + 2k
            = -3i + 5j + 2k
2.   Find CE
-   CE = CB + BA + AE
-   *BA=-OC and AE=OD
-   CE = 3j - (-i + j +2k) + 2i – j
            = i + 2i + 3j – j – 2k
            = 3i + j – 2k
3.   DB.CE = 0 (bisect=perpendicular to each other)
DB.CE = (-3i + 5j + 2k).( 3i + j – 2k)
            = (-3, 5, 2).(3, 1, -2)
            = (-3x3) + (5x1) + (2x-2)
            = -9 + 5 – 4
            ≠ 0 ??
                 

Q.2

C and D are points defined respectively by position vectors c and d. If |c| = 5, |d| = 7 and c.d= 4, find vector |CD|


Vectors
Q3.

Let A = (4, -3) and B = (7,1). Find N, such that vector AN = 3BN

[gta007]

Q2.
C and D are points defined respectively by position vectors c and d. If |c| = 5, |d| = 7 and c.d= 4, find vector |CD|

Firstly....







Use cosine rule to find vector |CD|

[gta007]

Q3.

Let A = (4, -3) and B = (7,1). Find N, such that vector AN = 3BN

Okay I try to picture it on a line like this......it's not mathematically correct but works for me:
A(4,-3) _________________B(7,1)_______N(x,y)









Equating coefficients of and

[gta007]

Q1.
(Image removed from quote.)
2.   Find CE
-   CE = CB + BA + AE
-   *BA=-OC and AE=OD
-   CE = 3j - (-i + j +2k) + 2i – j
            = i + 2i + 3j – j – 2k
            = 3i + j – 2k
3.   DB.CE = 0 (bisect=perpendicular to each other)
DB.CE = (-3i + 5j + 2k).( 3i + j – 2k)
            = (-3, 5, 2).(3, 1, -2)
            = (-3x3) + (5x1) + (2x-2)
            = -9 + 5 – 4
            ≠ 0 ??
               

I seem to get the same problem. At first I overlooked the -2k and had 2k.
Then DB.CE = 0, and then still got the correct answer of 69.71 in the end.

Dunno, me and your working out are the same, might need to wait for some of the maths whizzes to log on, and see how to correctly get the positive 2k.

[evaporade]

bisect means cut into halves

[Mao]

yes, you need to show that given the intersection point X, DX=XB and CX=XE
then using dot product, find the acute angle between them (there are two possible angles, one obtuse and one acute).