There's a Triangle with coordinate: (2,1) (6,3) (4,7). I need to find the area of the triangle using vectors.

Question

I have a POW due this friday (3/30) I need to find all sides and angles and again using vectors...any attempted answer would be much appreciated thanks.

In rectangular form

Answer 1

Let's label the points
A(2,1)
B(6,3)
C(4,7)

Now create two vectors u and v, from the points.

u = AB = <6-2,3-1> = <4,2>
v = AC = <4-2,7-1> = <2,6>

We can obtain the area of the triangle by cross multiplying the x and y components of the vectors u and v and taking one-half of the absolute value.

Area = (1/2)(4*6 - 2*2) = (1/2)(24 - 4) = (1/2)(20) = 10
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You can find sides with the Pythagorean Theorem. Just find the distance between the points. I will omit the details.

AB = √20 = 2√5
AC = √40 = 2√10
BC = √20 = 2√5

At this point we could note that the triangle in question is an isosceles right triangle. Its angles are 45°, 45°, and 90°.

But even if the results weren't so "nice" we could still calculate the angles as shown below.

The area of a triangle with two sides a, b, and the included angle C is given by:

Area = (1/2)ab*sinC

Plugging in we have:

Area = (1/2) ||AB|| ||AC|| sin C

10 = (1/2)(2√5)(2√10) sin C
10 = 2√50 sinC = 10√2 sin C

sin C = 1/√2
C = arcsin(1/√2) = π/4 = 45°

The remaining angles can be calculated by the Law of Sines.

AB / sinC = AC / sinB = BC / sinA

Answer 2

In polar form or rectangular form?