The goal of this item is to show that if u = (u, u) and lel...

  

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The goal of this item is to show that if u = (u₁, u₂) and lel to each other if and only if u₁v2 u2v₁ = 0. = (v1, v2) are vectors in R2, then, u and 7 are paral- (a) Begin by stating the definition of u and being parallel to each other (note: in our textbook/class slides....not the above statement!). (b) () Prove the forward implication: Show that if ū = (u₁, ₂) and 7 = (v₁, v₂) are parallel to each other, then u₁v2 − u2v₁ = 0. Hint: just use direct substitution using (a). (c) () Prove the backward implication: Show that if u₁v2 - U2v1 = 0, then ua for some a € R² or 7 = bu for some b E R². In other words, to reach this conclusion, you must be able to find the value of a or b. Hint: do a Case-by-Case Analysis with Case 1: u₁ # 0, and Case 2: u₁ = 0. Recall that 02 is parallel to all vectors in R2, and a non-zero number has a reciprocal (Recall the axioms of R). Case 2 will have sub- cases: Case 2a: u20, and Case 2b: u2 = 0. Be sure to explain carefully in Case 2a why (0, u₂) is parallel to (0, v₂). The goal of this item is to show that if u = (u₁, u₂) and lel to each other if and only if u₁v2 u2v₁ = 0. = (v1, v2) are vectors in R2, then, u and 7 are paral- (a) Begin by stating the definition of u and being parallel to each other (note: in our textbook/class slides....not the above statement!). (b) () Prove the forward implication: Show that if ū = (u₁, ₂) and 7 = (v₁, v₂) are parallel to each other, then u₁v2 − u2v₁ = 0. Hint: just use direct substitution using (a). (c) () Prove the backward implication: Show that if u₁v2 - U2v1 = 0, then ua for some a € R² or 7 = bu for some b E R². In other words, to reach this conclusion, you must be able to find the value of a or b. Hint: do a Case-by-Case Analysis with Case 1: u₁ # 0, and Case 2: u₁ = 0. Recall that 02 is parallel to all vectors in R2, and a non-zero number has a reciprocal (Recall the axioms of R). Case 2 will have sub- cases: Case 2a: u20, and Case 2b: u2 = 0. Be sure to explain carefully in Case 2a why (0, u₂) is parallel to (0, v₂).

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a Definition of vectors u and v being parallel to each other Two vectors u and v in R2 are parallel ... View the full answer