Question

find the cross product of r and f where f is i+j+k and r is the distance between two points whose coordinates are (-2,3,4) and (1,2,3)
please answer fast it is quit urgent.....​

Answer 1

Explanation:

The dot product is a multiplication of two vectors that results in a scalar. In this section, we introduce a product of two vectors that generates a third vector orthogonal to the first two. Consider how we might find such a vector. Let u=⟨u1,u2,u3⟩ and v=⟨v1,v2,v3⟩ be nonzero vectors. We want to find a vector w=⟨w1,w2,w3⟩ orthogonal to both u and v —that is, we want to find w such that u⋅w=0 and v⋅w=0. Therefore, w1, w2, and w3 must satisfy

u1w1+u2w2+u3w3v1w1+v2w2+v3w3==00.

If we multiply the top equation by v3 and the bottom equation by u3 and subtract, we can eliminate the variable w3, which gives

(u1v3−v1u3)w1+(u2v3−v2u3)w2=0.

If we select

w1w2==u2v3−u3v2−(u1v3−u3v1),

we get a possible solution vector. Substituting these values back into the original equations gives

w3=u1v2−u2v1.

That is, vector

w=⟨u2v3−u3v2,−(u1v3−u3v1),u1v2−u2v1⟩

is orthogonal to both u and v, which leads us to define the following operation, called the cross product.