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: $$\begin{array}{llllll} \text { (a } & \text { b } & \text { d) } .| & \text { (a } & \text { b } & \text { d) } &|. \text { (b } & \text { a } & \text { d) } \end{array}$$

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:
$$\begin{array}{llllll}
\text { (a } & \text { b } & \text { d) } .| & \text { (a } & \text { b } & \text { d) } &|. \text { (b } & \text { a } & \text { d) }
\end{array}$$

Key Concepts

Scalar Triple Product

The scalar triple product involves three vectors and is computed as the dot product of one vector with the cross product of the other two. This operation yields a scalar value that represents the signed volume of the parallelepiped defined by the three vectors. The sign of the result indicates the orientation of the three vectors, and if the scalar triple product equals zero, the vectors are coplanar.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It represents a projection of one vector onto another and is computed by multiplying the corresponding entries and summing these products. In geometric contexts, it is used to determine angles between vectors and to assess orthogonality.

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space, resulting in another vector that is perpendicular to the plane containing the original vectors. Its magnitude is equal to the area of the parallelogram spanned by the two vectors. The direction of the cross product is determined by the right-hand rule, making it central to problems involving orientations in right-handed coordinate systems.

Right-Handed Coordinate System

A right-handed coordinate system is a standard for defining the orientation of three-dimensional space. In this arrangement, the thumb, index, and middle fingers of the right hand used in sequence can represent the positive directions of the x-, y-, and z-axes respectively. This system underlies most conventional vector operations such as the cross product, ensuring consistency in the interpretation of directions and orientations.

Transcript

00:01 The triple scalar product of three vectors is shown as parentheses with the three vectors in there.

00:09 And what this is defined to be is it is defined to be a dotted into the cross product between the last two vectors.

00:22 However, it is also defined to be several other things.

00:26 So you may exchange the dot and the crossed.

00:33 So it is also a cross b dotted into d.

00:38 And it is also equal to the determinant with each row in that matrix for which you're finding the determinant occupied by one of the vectors.

00:54 So the first vector occupies the first row.

01:00 Second vector, the second row, et cetera.

01:04 And you know that the determinant just changes sign depending on how many rows or columns you've exchanged or swapped.

01:15 But we'll go ahead and use the determinant definition.

01:21 Can be a little bit quicker than the dot product could be, but it's a nicely organized matrix.

01:31 So we put our components of a, a, b, and d are shown as off to the left.

01:39 B is 3 minus 4 and 0, and d is 6 2 minus 3.

01:47 And those zeros are going to make things a little bit easier to deal with.

01:52 So there are really only two sub -determinants that we have to deal with.

01:56 The first comes from blocking out the first row...

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