Why is cosine used in dot products and sine used in cross products?

Which of the following trigonometric function is involved in cross product?Group of answer choicessinetangentcotangentcosine

If  and , then the cosine of the angle θ between the two vectors is:

Cosine similarity is a metric used to measure the similarity between two non-zero vectors in a multi-dimensional space. It's widely used in various fields, including natural language processing, information retrieval, recommendation systems, and more. Cosine similarity is particularly popular in text analysis and document retrieval tasks.

Triple Products Since the cross product of two vectors is itself a vector, it can be dotted or crossed with a third vector to form a triple product. (i) Scalar triple product: A · (B × C). Geometrically, |A · (B × C)| is the volume of the parallelepiped generated by A, B, and C, since |B × C| is the area of the base, and |A cos θ| is the altitude (Fig. 1.12). Evidently, A · (B × C) = B · (C × A) = C · (A × B), (1.15) for they all correspond to the same figure. Note that “alphabetical” order is preserved—in view of Eq. 1.6, the “nonalphabetical” triple products, A · (C × B) = B · (A × C) = C · (B × A), have the opposite sign. In component form, A · (B × C) = Ax Ay Az Bx By Bz Cx Cy Cz . (1.16) Note that the dot and cross can be interchanged: A · (B × C) = (A × B) · C (this follows immediately from Eq. 1.15); however, the placement of the parentheses is critical: (A · B) × C is a meaningless expression—you can’t make a cross product from a scalar and a vector. x y z n 1 2 3 FIGURE 1.11 B C A θ n FIGURE 1.12 8 Chapter 1 Vector Analysis (ii) Vector triple product: A × (B × C). The vector triple product can be simplified by the so-called BAC-CAB rule: A × (B × C) = B(A · C) − C(A · B). (1.17) Notice that (A × B) × C = −C × (A × B) = −A(B · C) + B(A · C) is an entirely different vector (cross-products are not associative). All higher vector products can be similarly reduced, often by repeated application of Eq. 1.17, so it is never necessary for an expression to contain more than one cross product in any term. For instance, (A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C); A × [B × (C × D)] = B[A · (C × D)] − (A · B)(C × D).

The Law of Cosines is used to solve triangles in which two sides and the including angle (SAS) are known or those in which three sides (SSS) are known.Group of answer choicesTrueFalse