Question

Let the four horizontal compass directions north, east, south, and west be represented by units vectors n^, e^, s^, and w^, respectively. Vertically up and down are represented as u^ and d^. Let us also identify unit vectors that are halfway between these directions, such as (ne)^ for northeast. Rank the magnitudes of the following cross products from the largest to the smallest. (Use only ">" or "=" symbols. Do not include any parentheses around the letters or symbols.)

(a) n^ x n^(b) w^ x (ne)^(c) u^ x (ne)^(d) n^ x (nw)^(e) n^ x e^

Answer 1

c = e > b = d > a

Step-by-step explanation:

Given vectors are all unit vectors, therefore they have a magnitude of 1

Let a, b be two vectors and magnitude of cross product of these two vectors is (magnitude of a) × (magnitude of b) × (sine of angle between these two vectors)

As all are unit vectors their magnitude is 1 and therefore in this case the cross product between any two vectors depends on the sine of angle between those two vectors

In option a as both the vectors are same, the angle between them will be zero and sin0° will also be 0

In option b angle between those two vectors is 135° and sin135° is 1 ÷ √2

In option c angle between those two vectors is 90° and sin90° is 1

In option d angle between those two vectors is 45° and sin45° is 1 ÷ √2

In option e angle between those two vectors is 90° and sin90° is 1

So by comparison of magnitudes of cross products in each option, the order will be c = e > b = d > a