Volume of a Parallelepiped Three vectors 𝐮, 𝐯, and 𝐰 are given. (a) Find their scalar triple pro

step 1 In this problem, we have to start off by finding the scalar triple product given the vector is U

Volume of a Parallelepiped Three vectors 𝐮, 𝐯, and 𝐰 are...

Volume of a Parallelepiped Three vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are given. (a) Find their scalar triple product $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) .$ (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$ \mathbf{u}=2 \mathbf{i}-2 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{v}=3 \mathbf{i}-\mathbf{j}-\mathbf{k}, \quad \mathbf{w}=6 \mathbf{i} $$

Volume of a Parallelepiped Three vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are given. (a) Find their scalar triple product $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) .$ (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine.
$$
\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{v}=3 \mathbf{i}-\mathbf{j}-\mathbf{k}, \quad \mathbf{w}=6 \mathbf{i}
$$

Key Concepts

Volume of a Parallelepiped

The volume of a parallelepiped defined by three vectors is given by the absolute value of their scalar triple product. This provides a direct way to compute the three-dimensional space enclosed by the vectors.

Scalar Triple Product

The scalar triple product is an operation on three vectors that involves taking the dot product of one vector with the cross product of the other two. Its absolute value represents the volume of the parallelepiped formed by the three vectors, and a zero value indicates that the vectors are coplanar.

Cross Product

The cross product of two vectors is a vector perpendicular to the plane containing the original vectors. Its magnitude equals the area of the parallelogram spanned by the two vectors and plays a key role in calculating the scalar triple product.

Dot Product

The dot product is a measure of the projection of one vector onto another and is computed as the product of their magnitudes and the cosine of the angle between them. It is used in the scalar triple product to combine with the cross product and yield a scalar quantity related to volume.

Coplanarity

Coplanarity refers to the condition where three or more vectors lie in the same plane. In the context of the scalar triple product, if the result is zero, it means the vectors are coplanar, and the volume of the parallelepiped they form is zero.

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