Which pair of vectors is orthogonal?
A. [tex]u =\langle-3,2\rangle[/tex] and [tex]v =\langle-2,1\rangle[/tex]
B. [tex]u =\langle 1,5\rangle[/tex] and [tex]v =\langle-5,1\rangle[/tex]
C. [tex]u =\langle 2,1)[/tex] and [tex]v =\langle-4,3\rangle[/tex]
D. [tex]u =\langle 2,3\rangle[/tex] and [tex]v =\langle 3,1\rangle[/tex]
Answer (2)
The pair of vectors that is orthogonal is Pair B: u = ⟨ 1 , 5 ⟩ and v = ⟨ − 5 , 1 ⟩ . Their dot product equals zero, confirming their orthogonality. Therefore, the correct answer is B.
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The problem asks to identify the orthogonal pair of vectors from the given options.
Recall that two vectors are orthogonal if their dot product is zero.
Calculate the dot product for each pair of vectors.
The pair u = ⟨ 1 , 5 ⟩ and v = ⟨ − 5 , 1 ⟩ has a dot product of 0, thus they are orthogonal. u = ⟨ 1 , 5 ⟩ and v = ⟨ − 5 , 1 ⟩
Explanation
Problem Analysis and Strategy We are given four pairs of vectors and asked to determine which pair is orthogonal. Two vectors are orthogonal if their dot product is zero. The dot product of two vectors u = ⟨ a , b ⟩ and v = ⟨ c , d ⟩ is given by u ⋅ v = a c + b d . We will calculate the dot product for each pair of vectors.
Calculate Dot Product of First Pair For the first pair, u = ⟨ − 3 , 2 ⟩ and v = ⟨ − 2 , 1 ⟩ , the dot product is: u ⋅ v = ( − 3 ) ( − 2 ) + ( 2 ) ( 1 ) = 6 + 2 = 8 Since the dot product is 8, these vectors are not orthogonal.
Calculate Dot Product of Second Pair For the second pair, u = ⟨ 1 , 5 ⟩ and v = ⟨ − 5 , 1 ⟩ , the dot product is: u ⋅ v = ( 1 ) ( − 5 ) + ( 5 ) ( 1 ) = − 5 + 5 = 0 Since the dot product is 0, these vectors are orthogonal.
Calculate Dot Product of Third Pair For the third pair, u = ⟨ 2 , 1 ⟩ and v = ⟨ − 4 , 3 ⟩ , the dot product is: u ⋅ v = ( 2 ) ( − 4 ) + ( 1 ) ( 3 ) = − 8 + 3 = − 5 Since the dot product is -5, these vectors are not orthogonal.
Calculate Dot Product of Fourth Pair For the fourth pair, u = ⟨ 2 , 3 ⟩ and v = ⟨ 3 , 1 ⟩ , the dot product is: u ⋅ v = ( 2 ) ( 3 ) + ( 3 ) ( 1 ) = 6 + 3 = 9 Since the dot product is 9, these vectors are not orthogonal.
Final Answer The pair of vectors with a dot product of zero is u = ⟨ 1 , 5 ⟩ and v = ⟨ − 5 , 1 ⟩ . Therefore, this pair of vectors is orthogonal.
Examples
Orthogonal vectors are fundamental in computer graphics for creating coordinate systems and defining spatial relationships. For example, in 3D modeling, orthogonal vectors are used to define the axes of a local coordinate system for an object, ensuring that transformations like rotations and scaling are applied correctly. This ensures that objects are rendered accurately and maintain their intended spatial relationships within a scene.
