show that the quadrilateral with vertices at the following points is a parallelogram and find its area. a = (5,2,0), b = (2,6,1), c = (2,4,7), d = (5,0,6)

thanks steve. u really help me out.. really appreciate it .. =)

To prove that the quadrilateral with the given vertices is a parallelogram, we can use the following two properties:

1. Opposite sides of a parallelogram are parallel.
2. Diagonals of a parallelogram bisect each other.

Let's go step by step to determine if the given quadrilateral is a parallelogram.

Step 1: Calculate the vectors of the sides.

The vectors of the sides can be found by subtracting the coordinates of the corresponding vertices. Let's calculate the vectors for the sides using points A, B, C, and D:

Vector AB = (2-5, 6-2, 1-0) = (-3, 4, 1)
Vector BC = (2-2, 4-6, 7-1) = (0, -2, 6)
Vector CD = (5-2, 0-4, 6-7) = (3, -4, -1)
Vector DA = (5-5, 2-0, 0-6) = (0, 2, -6)

Step 2: Check if opposite sides are parallel.

To determine if opposite sides are parallel, we need to check if the corresponding vectors are scalar multiples of each other.

AB and CD:
We can check whether the direction ratios of AB and CD are proportional.

Direction ratio of AB = (-3, 4, 1)
Direction ratio of CD = (3, -4, -1)

If the direction ratios of AB and CD are proportional, then AB and CD are parallel. In this case, we can see that the direction ratios are negatives of each other, so AB and CD are parallel.

BC and DA:
We can check whether the direction ratios of BC and DA are proportional.

Direction ratio of BC = (0, -2, 6)
Direction ratio of DA = (0, 2, -6)

Again, we see that the direction ratios are negatives of each other, so BC and DA are parallel.

Since both pairs of opposite sides are parallel, the quadrilateral formed by the given vertices is a parallelogram.

Step 3: Check if diagonals bisect each other.

To determine if the diagonals bisect each other, we need to find the midpoints of the diagonals.

The midpoint of AC can be found by averaging the coordinates of A and C:
Midpoint AC = [(5+2)/2, (2+4)/2, (0+7)/2] = (3.5, 3, 3.5)

The midpoint of BD can be found by averaging the coordinates of B and D:
Midpoint BD = [(2+5)/2, (6+0)/2, (1+6)/2] = (3.5, 3, 3.5)

Since both midpoints are equal, the diagonals AC and BD bisect each other.

Therefore, we have shown that the given quadrilateral is a parallelogram.

Step 4: Calculate the area of the parallelogram.

To find the area of the parallelogram, we can use the formula: Area = base × height.

In a parallelogram, any side can be considered the base, and the height is the perpendicular distance between the base and the opposite side.

Let's consider side AB as the base. The height of the parallelogram can be found by calculating the perpendicular distance between AB and side CD.

The equation of a line perpendicular to AB passing through C can be found using the cross product of AB and CD.

Cross product AB × CD = (-3, 4, 1) × (3, -4, -1)
= (-8, 8, -12)

The equation of the line perpendicular to AB passing through C is:
-8x + 8y - 12z + D = 0

Substituting the coordinates of point C(2, 4, 7), we can find the value of D:
-8(2) + 8(4) - 12(7) + D = 0
D = 40

So, the equation of the line perpendicular to AB passing through C is:
-8x + 8y - 12z + 40 = 0

Now, we can find the perpendicular distance between AB and CD, which is the height of the parallelogram. The distance can be calculated as the absolute value of the dot product of the normal vector (-8, 8, -12) and a point on AB, divided by the magnitude of the normal vector.

Let's take point A (5, 2, 0) on AB:
Distance = | (-8)(5) + (8)(2) + (-12)(0) + 40 | / √((-8)^2 + 8^2 + (-12)^2)
= | -40 + 16 + 40 | / √(64 + 64 + 144)
= | 16 | / √(272)
= 16 / 16.49
≈ 0.97

So, the height of the parallelogram is approximately 0.97.

Now, we can calculate the area using the base (AB) and the height:

Area = base × height
= ||AB|| × height
= √((-3)^2 + 4^2 + 1^2) × 0.97
= √26 × 0.97
≈ 4.99

Therefore, the area of the parallelogram formed by the given vertices is approximately 4.99 square units.

AB has direction (-3,4,1)

BC has direction (0,-2,6)
CD has direction (3,-4,-1)
DA has direction (0,-2,6)

So, AB || CD
and BC || AD

since the sides are parallel in pairs, they must have the same lengths in pairs (I trust you can verify this).

the area is just
|(b-a)x(d-a)|
where a,b,c,d are the vectors to the points A,B,C,D.