Question

The cross product of two Euclidean vectors is defined as a⋅b=∣a∣₂ ∣b∣₂ sin(θ)uₙ where - θ∈[0:π] is the angle between a and b in the plane that contains both of them. - uₙ

is a unit vector perpendicular to the plane that contains a and b. Using simple geometrical arguments and drawings explain the following identities
(a) i×j=k.

Answer 1

The cross product of the vectors i and j, represented as i × j, equals the vector k. This can be proven using simple geometrical arguments and drawings. The resulting vector has a magnitude of 1 and is parallel to the positive z-axis.

Step-by-step explanation:

According to the definition of the cross product, the cross product of the vectors i and j, written as i × j, equals the vector k. This can be proven using simple geometrical arguments and drawings. When we apply the cross product definition to the unit vectors i and j, we find that i × i - j × j = k × k = 0. All other cross products of these unit vectors must be vectors of unit magnitudes because i, j, and k are orthogonal. For the pair i and j, the magnitude is |i × j| = ij sin 90° = (1)(1)(1) = 1. The direction of the vector product i × j must be along the positive z-axis, which is represented by the vector k.