Vector Geometry: Quadrangular Pyramid with Inner and Cross Products

[devinaxxx]

Homework Statement

A quadrangular pyramid OABCD with square ABCD as the bottom. OA = 1, AB = 2, BC = 2 Also, OA perpendicular to AB, OA perpendicular to AD.
Question 1 : Find the inner product $\overrightarrow {OA}.\overrightarrow {OB}$ and the size of the cross product |$\overrightarrow {OA}X\overrightarrow {OB}$ |

2. Let E denote the point dividing the OD into 2: 3, and let F be the midpoint of OC. **Also A plane including three points A, E, and F and a point intersecting the side OB or its extension are defined as G**. At this time, express OG with OA,OB,and OC . can someone give me hint? thanks

Homework Equations

The Attempt at a Solution

I got the first question that the inner product is OA.OB=1
but the second question,

I don't understand where is G in the plane and what is the relation with A,E,F?
And why $\overrightarrow {AG}= s. \overrightarrow {AE}+t. \overrightarrow {AF}.\overrightarrow {OB}$ ??

 

[andrewkirk]

I suspect this problem is most easily solved using coordinate geometry. Set
A=(0,0,0)
B=(0,2,0)
C=(2,2,0)
D=(2,0,0)
From part 1 of the question we have O=(0,0,1).

Part 2 of the problem is stated rather unclearly. It seems to be missing key phrases. But my best guess is that it wants you to do the following steps:

1. calculate the coordinates of F as (O+C)/2
2. calculate the coordinates of E as (3O+2D)/(3+2)
3. find the equation of the unique plane through points A, E and F
4. find the equation of the unique line that passes through O and B
5. find the coordinates of the point G that is the intersection of the plane from step 3 and the line from step 4