Question Video: Solving Problems Involving Parallel and Perpendicular Vectors in 2D Mathematics

If 𝐀 = ⟨𝑘², 1⟩, 𝐁 = ⟨2, −8⟩, and 𝐀 ⊥ 𝐁, then 𝑘 = ＿. [A] 2 [B] −2 [C] 2, −2 [D] 4

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Video Transcript

If vector 𝐀 is equal to 𝑘 squared, one; vector 𝐁 is equal to two, negative eight; and vector 𝐀 is perpendicular to vector 𝐁, then 𝑘 equals what. Is it (A) two, (B) negative two, (C) two or negative two, or (D) four?

The notation in the question tells us that vectors 𝐀 and 𝐁 are perpendicular. We know that for any two perpendicular vectors 𝐕 sub one and 𝐕 sub two, their dot or scalar product is equal to zero. We recall that we calculate the dot product of any two vectors by multiplying their corresponding components and then finding the sum of these values. This gives us a scalar quantity.

In this question, we need to calculate the dot product of the vectors 𝑘 squared, one and two, negative eight. This is equal to 𝑘 squared multiplied by two plus one multiplied by negative eight. This simplifies to two 𝑘 squared minus eight. As the vectors are perpendicular, we know that this equals zero. We can then solve this equation by firstly adding eight to both sides. Two 𝑘 squared is, therefore, equal to eight. Dividing through by two, we see that 𝑘 squared is equal to four. We can then square root both sides of our equation. And recalling that our answer can be positive or negative, we have 𝑘 is equal to positive or negative root four. We know that the square root of four is equal to two. Therefore, 𝑘 is equal to positive or negative two.

This means that the correct answer to this question is option (C). If 𝐀 is equal to 𝑘 squared, one and 𝐁 is equal to two, negative eight where vectors 𝐀 and 𝐁 are perpendicular, then 𝑘 can be equal to two or negative two. We can check this answer by substituting our values of 𝑘 into vector 𝐀. We know that two squared and negative two squared are both equal to four. Therefore, vector 𝐀 is equal to four, one. As the dot product of four, one and two, negative eight is equal to zero, we know that our answer is correct.