The question asks me to find the area of the triangle with the given vertices. (The area A of the triangle having u and v as adjacent sides is given by A=1/2 ||u x v||.)
the vertices are:
(0,0,0) (1,2,3) (-3,0,0)
so, call the 3 points a,b,c
u=b-a = (1,2,3)
v=c-a = (-3,0,0)
now use the determinant to figure ||u x v||
Thank you!
To find the area of the triangle with the given vertices, we can use the formula A = 1/2 ||u x v||, where u and v are the vectors formed by two sides of the triangle and ||u x v|| represents the magnitude, or length, of the cross product of u and v.
Step 1: Define the vectors u and v.
Let's use the given vertices to define the vectors u and v. The first two vertices will be used to find vector u, and the first and third vertices will be used to find vector v. We can calculate these vectors by subtracting the coordinates of the vertices.
u = (1, 2, 3) - (0, 0, 0)
u = (1, 2, 3)
v = (-3, 0, 0) - (0, 0, 0)
v = (-3, 0, 0)
Step 2: Calculate the cross product of u and v.
The cross product of two vectors is a vector that is perpendicular to both of the original vectors. We can calculate the cross product of u and v using the following formula:
u x v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1)
Plugging in the values:
u x v = (2 * 0 - 3 * 0, 3 * (-3) - 1 * 0, 1 * 0 - 2 * (-3))
u x v = (0, -9, 6)
Step 3: Calculate the magnitude ||u x v||.
The magnitude, or length, of the cross product vector ||u x v|| can be found using the following formula:
||u x v|| = sqrt((u x v) · (u x v))
Plugging in the values:
||u x v|| = sqrt((0)^2 + (-9)^2 + 6^2)
||u x v|| = sqrt(0 + 81 + 36)
||u x v|| = sqrt(117)
||u x v|| ≈ 10.82
Step 4: Calculate the area of the triangle.
Now that we have the magnitude of the cross product, we can use the formula A = 1/2 ||u x v|| to find the area of the triangle.
A = 1/2 * ||u x v||
A = 1/2 * 10.82
A ≈ 5.41
Therefore, the area of the triangle with the given vertices is approximately 5.41 square units.
To find the area of a triangle given its vertices, we can use the formula provided, A = 1/2 ||u x v||, where u and v are the vectors representing two adjacent sides of the triangle. Here's how you can solve this:
Step 1: Find the vectors representing the sides of the triangle.
- The first vector, u, is the difference between the second vertex and the first vertex:
u = (1, 2, 3) - (0, 0, 0) = (1, 2, 3).
- The second vector, v, is the difference between the third vertex and the first vertex:
v = (-3, 0, 0) - (0, 0, 0) = (-3, 0, 0).
Step 2: Calculate the cross product of the two vectors, u x v.
- The cross product can be found by taking the determinants of the following matrix:
| i j k |
| 1 2 3 |
| -3 0 0 |
- Evaluate the determinants:
i-component = (2 * 0) - (3 * 0) = 0,
j-component = -(1 * 0) - (3 * -3) = 9,
k-component = (1 * 0) - (2 * -3) = 6.
- Combine the components to get the cross product: u x v = (0, 9, 6).
Step 3: Calculate the magnitude of the cross product, ||u x v||.
- The magnitude can be calculated using the formula: ||u x v|| = sqrt(x^2 + y^2 + z^2)
- Substitute the values: ||u x v|| = sqrt(0^2 + 9^2 + 6^2) = sqrt(0 + 81 + 36) = sqrt(117) ≈ 10.82.
Step 4: Substitute the magnitude into the formula A = 1/2 ||u x v|| to find the area.
- Substitute the values: A = 1/2 * 10.82 = 5.41 square units(approximately).
Therefore, the area of the triangle with the given vertices is approximately 5.41 square units.