Video: Solving Problems Using the Dot Product

Given the vectors 𝚨 = [π‘₯ and βˆ’4] and 𝚩 = [π‘₯ and π‘₯ βˆ’ 1] are perpendicular, find the value of π‘₯.

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Video Transcript

Given the vectors 𝚨 π‘₯, negative four and 𝚩 π‘₯, π‘₯ minus one are perpendicular, find the value of π‘₯.

We know that two vectors are perpendicular if their dot or scalar product is equal to zero. We find the dot product of two vectors by multiplying their π‘₯-components and 𝑦-components. We then find the sum of these values. In this question, the π‘₯-components of vectors 𝚨 and 𝚩 are both equal to π‘₯. Multiplying π‘₯ by π‘₯ gives us π‘₯ squared. The 𝑦-components are equal to negative four and π‘₯ minus one, so we need to multiply negative four by π‘₯ minus one. We can do this by distributing our parentheses. We multiply negative four by π‘₯ and negative four by negative one. Our expression simplifies to π‘₯ squared minus four π‘₯ plus four. As the vectors are perpendicular, we know that this is equal to zero.

We now have a quadratic equation that we can solve by factoring. The first term in both of the parentheses will be equal to π‘₯. To find the second term, we need to find a pair of numbers that sum to negative four and have a product of positive four. Negative two multiplied by negative two is equal to positive four, and the sum of these values is negative four. As both of the parentheses are identical, the only solution will be when π‘₯ minus two is equal to zero. Adding two to both sides of this equation gives us a value of π‘₯ equal to two.

We can check this answer by substituting our value into the initial vectors. Vector 𝚨 is equal to two, negative four. Vector 𝚩 is equal to two, one. The dot product of these vectors is equal to four plus negative four as two multiplied by two is four and negative four multiplied by one is negative four. As this is equal to zero, we have confirmed that the two vectors are indeed perpendicular when π‘₯ is equal to two.

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