Use the cross product to find the area of the triangle with vertices:
P=(1,1,5)
Q=(3,4,3)
R=(1,5,7)
See:
http://www.jiskha.com/display.cgi?id=1283403006
Oh triangles, they're always trying to pull you in different directions! Let's calculate the area using the cross product.
First, let's get vectors PQ and PR.
PQ = Q - P
= (3, 4, 3) - (1, 1, 5)
= (2, 3, -2)
PR = R - P
= (1, 5, 7) - (1, 1, 5)
= (0, 4, 2)
Now, let's find the cross product of PQ and PR.
PQ x PR = (2, 3, -2) x (0, 4, 2)
To calculate the cross product:
i (component) = (3 * 2) - (-2 * 4) = 14
j (component) = (-2 * 0) - (2 * 4) = -12
k (component) = (2 * 4) - (3 * 0) = 8
So, PQ x PR = (14, -12, 8).
Lastly, let's find the magnitude of the cross product.
|PQ x PR| = √(14^2 + (-12)^2 + 8^2)
= √(196 + 144 + 64)
= √404
≈ 20.1
Therefore, the area of the triangle with vertices P, Q, and R is approximately 20.1 square units. Now that's one tri-angled joke!
To find the area of a triangle using the cross product, follow these steps:
Step 1: Find the vectors PQ and PR.
PQ = Q - P = (3, 4, 3) - (1, 1, 5) = (2, 3, -2)
PR = R - P = (1, 5, 7) - (1, 1, 5) = (0, 4, 2)
Step 2: Take the cross product of vectors PQ and PR.
Cross product of PQ and PR = PQ x PR
= (2, 3, -2) x (0, 4, 2)
= ((3*(-2)) - (4*(-2)), ((2*2) - (-2*0)), ((2*4) - (3*0)))
= (-6 + 8, 4 + 0, 8 - 0)
= (2, 4, 8)
Step 3: Find the magnitude of the cross product.
Magnitude of the cross product = |PQ x PR| = √(2^2+4^2+8^2) = √(4+16+64) = √84 = 2√21
Step 4: Divide the magnitude by 2 to find the area of the triangle.
Area = (1/2) * Magnitude of the cross product = (1/2) * 2√21 = √21
Therefore, the area of the triangle with vertices P=(1,1,5), Q=(3,4,3), and R=(1,5,7) is √21.
To find the area of a triangle using the cross product, we need to calculate the cross product of two sides of the triangle.
Step 1: Find two vectors representing two sides of the triangle
To find two sides of the triangle, we can use the vertices P, Q, and R. Let's take the sides PR and PQ.
Side PR can be represented by the vector PR = R - P:
PR = (1, 5, 7) - (1, 1, 5) = (0, 4, 2)
Side PQ can be represented by the vector PQ = Q - P:
PQ = (3, 4, 3) - (1, 1, 5) = (2, 3, -2)
Step 2: Calculate the cross product of the two vectors
The cross product of two vectors A = (a1, a2, a3) and B = (b1, b2, b3) is given by:
A x B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)
Calculating the cross product of PR and PQ, we have:
PR x PQ = (4(-2) - 2(3), 2(2) - 0(-2), 0(3) - 4(2))
= (-8 - 6, 4, -8)
= (-14, 4, -8)
Step 3: Calculate the magnitude of the cross product vector
The magnitude (or length) of a vector V = (x, y, z) is given by:
|V| = sqrt(x^2 + y^2 + z^2)
Calculating the magnitude of the cross product vector PR x PQ, we have:
|PR x PQ| = sqrt((-14)^2 + 4^2 + (-8)^2)
= sqrt(196 + 16 + 64)
= sqrt(276)
= 16.61 (rounded to two decimal places)
Step 4: Calculate the area of the triangle
The area of the triangle can be found using the formula:
Area = 1/2 * |PR x PQ|
Substituting the magnitude of the cross product vector we calculated:
Area = 1/2 * 16.61
= 8.30 square units (rounded to two decimal places)
Therefore, the area of the triangle with vertices P, Q, and R is approximately 8.30 square units.