If a, b, c are unit vectors such that a + b + c = 0, then the value of a.b + b.c + c.a isReview Later13-3/2None of the Above

The volume of the  parallelepiped determined by the vectors a=(1,2,3), b=(-1, 1, 2) and c=(2, 1, 4) Question 12Answera.18 cubic unitsb.9 cubic unitsc.15/2 cubic unitsd.9 sq. units

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

If a:b = 2:3 and b:c = 4:5, then the ration a+b:b+c = ?

Let⃗a = ˆi − ˆj + √2ˆk and⃗ b = −12ˆi + 5ˆja) Compute 3⃗a −⃗ bb) Find the unit vector in the direction of⃗ b.c) Compute the magnitude of⃗a .d) Find the direction angles (using the direction cosines) for⃗a .

Let  ,  and  be three given vectors. If  is a vector such that  and  then  is equal to