Given the vectors 𝚨 𝑥, negative
four and 𝚩 𝑥, 𝑥 minus one are perpendicular, find the value of 𝑥.
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 𝑥-components and 𝑦-components. We then find the sum of these
values. In this question, the 𝑥-components
of vectors 𝚨 and 𝚩 are both equal to 𝑥. Multiplying 𝑥 by 𝑥 gives us 𝑥
squared. The 𝑦-components are equal to
negative four and 𝑥 minus one, so we need to multiply negative four by 𝑥 minus
one. We can do this by distributing our
parentheses. We multiply negative four by 𝑥 and
negative four by negative one. Our expression simplifies to 𝑥
squared minus four 𝑥 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 𝑥. 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 𝑥 minus two is equal to zero. Adding two to both sides of this
equation gives us a value of 𝑥 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 𝑥 is equal to two.