I can do A and B however C is confusing. (This is for a linear Algebra class) 1.

Question

I can do A and B however C is confusing. (This is for a linear Algebra class)

1. The goal of this item is to show that if" = 112) and v = v, v2) are vectors in R2, then, i and V are parallel to each other if and only if uiv2 -2v0 a. Begin by stating the definition of i and v being parallel to each other. b. (-) Prove the forward implication: Show that if" =11, 112) and (v, v2) are parallel to each other, then uv-2vi0. Hint: just use direct substitution using (-) Prove the backward implication: Show that ifu1V2-1121,1-0, then some a e R. In other words, to reach this conclusion, you must be able to find the value of . c· for Hint: do a Case-by-Case Analysis with Case 1: "I # 0, and Case 2: "1-0. Recall that 02 is parallel to all vectors in R2, and a non-zero number has a reciprocal. Case 2 will have sub-cases: Case 2a: 1120, and Case 2b: 112-0. Be sure to explain carefully in Case 2a why(0, u2) is parallel to 0, v2).

Explanation / Answer

a) The two vectors are said to be parallel, if they are having an angle of either 0 degree between them.

u being parallel to v means that u=av, where a is a constant

Since the angle between then is zero, which implies the cross product of two vector will be zero since

|uXv| = |u||v|sin(theta) n (where n is the vector in the crossed direction)

b)

Since u and v are parallel to each other, hence we can say that the angle between them will be zero.

The cross product of uXv = u1v2 - u2v1

Since the angle is zero degrees, hence cross product is zero, which implies that

u1v2 = u2v1

Hence proved

c)

It is given that u1v2 - v2u1 = 0

From the definition of cross product,

uXv = |u||v| sin(theta) n (where n is the vector in the crossed direction)

Since u1v2 - v2u1 = 0, it implies that

|u||v|sin(theta) = 0

There are two cases now, either modulus of |u| or |v| is zero or sin(theta) = 0

Case 1: sin(theta) = 0

It implies that the angle between the two vectors is either 0 degree or 180 degree, which means either the vector are parallel or anit-parallel

So we can write the vector u in terms of other vector v

where u = av, where a can be any value belonging to R

Case 2: When either of |u| or |v| is zero

This implies that the vector is having modulus of zero, which means u = (u1,u2) and v= (v1,v2)

If both the entries of the two dimensional vector is zero, which implies the vector is zero vector

which means it is parallel to every vector, since the cross product (u1v2 -u2v1) will always be zero

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