How Do You Calculate Vectors and Areas in Geometry Problems?

[custer]

i have no idea how to solve these 2 questions.. please direct me to the right way.

1. Find the vector from the origin to the center of mass of a thin triangular plate (uniform density) whose vertices are A(7, 7, 2), B(7, 4, 6), and C(4, 10, 1).

2. Find the area of the parallelogram determined by the points P(7, -5, 5), Q(4, 7, -3), R(-2,6,-4) and S(-5,18,-12)

 

[HallsofIvy]

i have no idea how to solve these 2 questions.. please direct me to the right way.

1. Find the vector from the origin to the center of mass of a thin triangular plate (uniform density) whose vertices are A(7, 7, 2), B(7, 4, 6), and C(4, 10, 1).

First find the center of mass: hint in a triangle with uniform density, just average the coordinates of the vertices. (That doesn't work for a figure with more than three vertices.)

2. Find the area of the parallelogram determined by the points P(7, -5, 5), Q(4, 7, -3), R(-2,6,-4) and S(-5,18,-12)

Do you know the formula for area of a parallelogram? You can use that.

Simpler is a formula from calculus: if $\vec{u}$ and $\vec{v}$ are vectors forming two adjacent sides of a parallelogram its area is $|\vec{u}\times\vec{v}|$.

Now, do something yourself before expecting any further help!