With respect to right-handed Cartesian coordinates, let...

With respect to right-handed Cartesian coordinates, let $a-[1,2,0], b-[3,-4,0], c-[3,5,2], d=[6,2,-3]$
Showing details, find:
$$\left.\begin{array}{llllll}
(\mathbf{i} & \mathbf{j} & \mathbf{k})\quad . & (\mathbf{i} & \mathbf{k} & \mathbf{j}
\end{array}\right)$$

Key Concepts

Cartesian Coordinate System

A Cartesian coordinate system is a framework for describing points in space using ordered sets of numbers (coordinates). In three?dimensional space, it typically involves three mutually perpendicular axes which are labeled with unit vectors. This system provides a basis for defining positions, directions, and operations such as dot and cross products in Euclidean space.

Standard Unit Vectors

The standard unit vectors, usually denoted as i, j, and k, represent the basis directions along the x?, y?, and z?axes respectively. They have unit length and are orthogonal to one another, forming the building blocks for more complex vector expressions and operations in three dimensions.

Right?Handed Orientation

A right?handed coordinate system is one in which the orientation of the axes follows the right?hand rule, meaning that if the thumb, index finger, and middle finger of the right hand are arranged mutually perpendicular, they align with the i, j, and k directions respectively. This notion of orientation is crucial when discussing operations like cross products and determinants, where the ordering of vectors directly influences the sign of the result.

Vector Permutations and Determinants

Permuting the order of basis vectors in an ordered triple, such as changing (i, j, k) to (i, k, j), is equivalent to applying a permutation to the coordinate system. Each permutation has a parity (even or odd), which affects the sign of determinants and related scalar quantities. Specifically, swapping two vectors (an odd permutation) reverses the orientation, which is reflected as a negative sign in the corresponding determinant or scalar triple product.

Scalar Triple Product

The scalar triple product, defined as the dot product of one vector with the cross product of the other two, yields the volume of the parallelepiped spanned by the three vectors, and its sign indicates the orientation (right?handed or left?handed) of the system. In problems involving different orderings of the standard unit vectors, the scalar triple product can be used indirectly to understand the effects of swapping vectors on the overall sign, often resulting in factors such as +1 or -1 reflecting the underlying permutation parity.

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