Find the volume of a parallelepiped with 3 edges defined by Vectore a = (-2, 1, 4), Vector b = (5, 9, 0) and Vectore c = (0, 3, -7).
Recall that a•b×c is just what you want.
So, v = (-2,1,4) • (-65,35,15) = 225
To find the volume of a parallelepiped defined by three vectors, you can use the scalar triple product.
The scalar triple product of three vectors a, b, and c is defined as:
V = a · (b x c)
Where a · (b x c) denotes the dot product of vector a with the cross product of vector b and vector c.
Let's calculate the volume step-by-step:
1. Calculate the cross product of vectors b and c:
b x c = | i j k |
| 5 9 0 |
| 0 3 -7 |
= (9 * -(-7) - 0 * 3) i - (5 * -(-7) - 0 * 0) j + (5 * 3 - 9 * 0) k
= (63 - 0) i - (35 - 0) j + (15 - 0) k
= 63i - 35j + 15k
2. Take the dot product of vector a with the cross product of vectors b and c:
a · (b x c) = (-2 * 63) + (1 * (-35)) + (4 * 15)
= -126 - 35 + 60
= -101
3. Take the absolute value of the result to obtain the volume:
V = | -101 |
= 101
The volume of the parallelepiped defined by vectors a, b, and c is 101.
To find the volume of a parallelepiped defined by three vectors, use the scalar triple product. Here are the steps to find the volume:
1. Start by calculating the scalar triple product of the three vectors: a, b, and c.
Scalar triple product = a · (b × c)
2. Find the cross product of vectors b and c.
b × c = (b.y * c.z - b.z * c.y, b.z * c.x - b.x * c.z, b.x * c.y - b.y * c.x)
b × c = (9 * (-7) - 0 * 3, 0 * 0 - 5 * (-7), 5 * 3 - 9 * 0)
Simplifying, we get: b × c = (-63, 35, 15)
3. Now, calculate the dot product of vector a with the cross product of b and c.
a · (b × c) = a.x * (b × c).x + a.y * (b × c).y + a.z * (b × c).z
Substituting the values, we get:
a · (b × c) = (-2 * -63) + (1 * 35) + (4 * 15)
Simplifying, we get: a · (b × c) = 211
4. The scalar triple product of the three vectors, a · (b × c), gives us the volume of the parallelepiped.
Therefore, the volume = 211.
So, the volume of the parallelepiped with edges defined by vectors a = (-2, 1, 4), b = (5, 9, 0), and c = (0, 3, -7) is 211 cubic units.