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Fill in the blank: If 𝐀 and 𝐁 are two perpendicular vectors and vector 𝐀 is equal to two, two, negative six, then vector 𝐁 may be equal to what.
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Is it (A) four, four, negative 12; (B) three, negative three, two; (C) negative three, negative three, two; (D) negative three, two, three; or (E) three, three, two?
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We are told in this question that the two vectors are perpendicular.
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And we recall that if two vectors 𝐮 and 𝐯 are perpendicular, their dot or scalar product is equal to zero.
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In this question, we’ll need to find the dot product of vector 𝐀 and the five options to see which one equals zero.
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Let’s begin with option (A).
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We calculate the dot product of any two vectors by multiplying the corresponding components and then finding the sum of these values.
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For option (A), we have two multiplied by four plus two multiplied by four plus negative six multiplied by negative 12.
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This simplifies to eight plus eight plus 72, which is equal to 88.
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As this is not equal to zero, option (A) is not correct.
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Let’s now consider option (B), the vector three, negative three, two.
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This time, the dot product is equal to two multiplied by three plus two multiplied by negative three plus negative six multiplied by two.
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This is equal to six plus negative six plus negative 12, which simplifies to negative 12 and is once again not equal to zero.
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Option (B) is not perpendicular to vector 𝐀.
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We repeat this process for option (C), giving us two multiplied by negative three plus two multiplied by negative three plus negative six multiplied by two.
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This is equal to negative six plus negative six plus negative 12, which is equal to negative 24 and once again is not equal to zero.
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We can therefore rule out option (C).
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Option (D) is the vector negative three, two, three.
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This time, the dot product is equal to two multiplied by negative three plus two multiplied by two plus negative six multiplied by three.
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This is equal to negative six plus four plus negative 18.
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This simplifies to negative 20, which once again is not equal to zero.
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Our final option (E) is the vector three, three, two.
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This time, the dot product is equal to two multiplied by three plus two multiplied by three plus negative six multiplied by two.
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This is equal to six plus six plus negative 12.
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Six plus six equals 12, and adding negative 12 to this gives us zero.
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This means that the dot product of vectors two, two, negative six and three, three, two is equal to zero.
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We can therefore conclude that out of the five options given, the vector that is perpendicular to vector 𝐀 is three, three, two.