Find the vectors' orthogonality, parallelism, or lack thereof. u = ⟨-3, 9, 6⟩, v = ⟨4, -12, -8⟩

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Find the vectors&#039; orthogonality, parallelism, or lack thereof. u = ⟨-3, 9, 6⟩, v = ⟨4, -12, -8⟩

nitruraviX · Accepted Answer

First, we find a dot product of the vectors u→ and v→  u→⋅v→=−3⋅4+9(−12)+6(−8)  =−12−108−48  =−168≠0  Second, we find the agle between the given vectors, for which we have to find length of vectors u→ and v→  |a→|=(−3)2+92+62  =9+81+36  =126  |b→|=42+(−12)2+(−8)2  =16+144+64  =224  So,  θ=arccos⁡(−168126224)  =arccos⁡(−1)  =180∘  =π  Therefore, the given vectors are parallel.  Result:  The given vectors are parallel

user_27qwe · Accepted Answer

To determine the orthogonality, parallelism, or lack thereof between the vectors 𝐮=⟨−3,9,6⟩ and 𝐯=⟨4,−12,−8⟩, we can use the dot product.The dot product of two vectors 𝐮 and 𝐯 is given by:𝐮·𝐯=u1v1+u2v2+u3v3where u1,u2,u3 are the components of vector 𝐮 and v1,v2,v3 are the components of vector 𝐯.Calculating the dot product for the given vectors, we have:𝐮·𝐯=(−3)(4)+(9)(−12)+(6)(−8)=−12−108−48=−168To determine the orthogonality between the vectors, we check if the dot product is equal to zero. In this case, 𝐮·𝐯=−168≠0, so the vectors 𝐮 and 𝐯 are not orthogonal.To determine the parallelism between the vectors, we check if the dot product is equal to the product of their magnitudes. If the dot product equals the product of the magnitudes (with the same sign), the vectors are parallel. In this case, we have:|𝐮|=(−3)2+92+62=126≈11.23|𝐯|=42+(−12)2+(−8)2=224≈14.97|𝐮|·|𝐯|≈11.23·14.97≈168.02Since 𝐮·𝐯=−168≠168.02, the vectors 𝐮 and 𝐯 are not parallel.Therefore, the vectors 𝐮 and 𝐯 are neither orthogonal nor parallel.

alenahelenash · Accepted Answer

Answer:- The vectors 𝐮 and 𝐯 are not orthogonal (𝐮·𝐯=−168≠0).- The vectors 𝐮 and 𝐯 are parallel (direction ratios are proportional).Explanation:The dot product of two vectors 𝐮 and 𝐯 is given by the formula:𝐮·𝐯=u1v1+u2v2+u3v3Substituting the values, we have:𝐮·𝐯=(−3)(4)+(9)(−12)+(6)(−8)Calculating this expression:𝐮·𝐯=−12−108−48𝐮·𝐯=−168Since the dot product 𝐮·𝐯 is not equal to zero, the vectors 𝐮 and 𝐯 are not orthogonal.To determine if the vectors are parallel, we can compare their direction ratios. If the direction ratios of 𝐮 and 𝐯 are proportional, the vectors are parallel.The direction ratios of 𝐮 are given by (−34,9−12,6−8), which simplifies to (−34,−34,−34).The direction ratios of 𝐯 are given by (4−3,−129,−86), which simplifies to (−43,−43,−43).Comparing the direction ratios, we can see that they are proportional. Therefore, the vectors 𝐮 and 𝐯 are parallel.

karton · Accepted Answer

Step 1:The dot product of two vectors 𝐮 and 𝐯 is given by the formula:𝐮·𝐯=u1·v1+u2·v2+u3·v3where u1,u2,u3 are the components of 𝐮 and v1,v2,v3 are the components of 𝐯.Let&#039;s calculate the dot product:𝐮·𝐯=(−3·4)+(9·−12)+(6·−8)Simplifying this expression, we get:𝐮·𝐯=−12−108−48Therefore:𝐮·𝐯=−168Step 2:Now, let&#039;s determine the orthogonality, parallelism, or lack thereof based on the dot product:1. If 𝐮·𝐯=0, the vectors are orthogonal.2. If 𝐮·𝐯≠0 and the magnitude of 𝐮·𝐯 is equal to the product of the magnitudes of 𝐮 and 𝐯, the vectors are parallel.3. If 𝐮·𝐯≠0 and the magnitude of 𝐮·𝐯 is not equal to the product of the magnitudes of 𝐮 and 𝐯, the vectors are neither orthogonal nor parallel.In this case, since 𝐮·𝐯=−168 and 𝐮·𝐯≠0, the vectors 𝐮 and 𝐯 are neither orthogonal nor parallel.

Determine whether the given vectors are orthogonal, parallel, or neither. u = ⟨-3, 9, 6⟩, v = ⟨4, -12, -8⟩

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