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).

For any three given vectors a, b, c ∈ R3 one can define the vectorsa′ = b × c(a, b, c), b′ = c × a(a, b, c), c′ = a × b(a, b, c).Here (a, b, c) = a · (b × c) stays for the triple/box product of a, b andc. The set of vectors {a′, b′, c′} is called reciprocal to the set of vectors{a, b, c}.Show that, if a, b and c 6 = 0, then(i) a · a′ = b · b′ = c · c′ = 1.[4 marks](ii) a′ · b = a′ · c = 0, b′ · a = b′ · c = 0 and c′ · a = c′ · b = 0.[6 marks]2. (a) Prove the following identity:∇ × (φA) = (∇φ) × A + φ(∇ × A),for any given vector field A(r) and a scalar field φ(r).[10 marks](b) Find a parametric equation for the semicirclex2 + y2 = a2, y > 0,using as a parameter the slope t = dy/dx of the tangent to the curve at(x, y).[10 marks]2

The cross product of two vectors A and B is another vectorGroup of answer choicesTrueFalse

Evaluate the cross product A⃗ ×B⃗ 𝐴→×𝐵→ of the two vectors above. They are separated by an angle of 150.° and have magnitudes given by:    Vector A: 9.00    Vector B: 6.00

A protester carries his sign of protest, starting from the origin of a 3D-Cartesian coordinate system,with the xy plane horizontal. He moves 40 m in the negative direction of the x-axis, then 20 m alonga perpendicular path to his left, and then 25 m up a water tower.(a) What does the scalar product of a vector, with itself, give? What about the vector product of avector with itself? Explain using no equations. (1)(b) In unit-vector notation, what is the displacement of the sign from start to end? (2)(c) The sign then falls to the foot of the tower. What is the magnitude of the displacement of thesign from the start to this new end? (3)(d) Verify the anti-commutative property of cross multiplication with the 1st and 2nd displacementvectors you got from (b) and (c), respectively.

Consider a vector a=[a1,a2]. Let's say that we compute the dot product of the vector with itself, i.e., compute a.a. What is the relationship between a.a and the magnitude of the vector ||a||?a.a = ||a||a.a = ||a||2a.a = √||a||