Fill in the blank: Given that vector 𝐀 equals one, three, 𝑥 squared and vector 𝐁 equals two, six, two, if vector 𝐀 is parallel to vector 𝐁, then 𝑥 equals what.
We begin by recalling that if two vectors 𝐮 and 𝐯 are parallel, then 𝐯 is equal to 𝑘 multiplied by 𝐮, where 𝑘 is a scalar constant. In this question, vector 𝐁 is therefore equal to 𝑘 multiplied by vector 𝐀. Two, six, two is equal to 𝑘 multiplied by one, three, 𝑥 squared. We can multiply a vector by a scalar by multiplying each of the components of the vector by that scalar. The right-hand side becomes 𝑘, three 𝑘, 𝑘𝑥 squared.
As this is equal to two, six, two, we know that the individual components must be equal. Firstly, we have two is equal to 𝑘. Next, we have six is equal to three 𝑘. And dividing both sides of this equation by three once again gives us 𝑘 is equal to two. Finally, we have two is equal to 𝑘𝑥 squared. As we have already found that 𝑘 is equal to two, this can be rewritten as two is equal to two 𝑥 squared. Dividing both sides of this equation by two gives us 𝑥 squared is equal to one. As we are looking to find the value of 𝑥, we can then square root both sides of this equation. The square root of one is one. Therefore, 𝑥 is equal to positive or negative one. If the vectors 𝐀 and 𝐁 are parallel, then 𝑥 is equal to positive one or negative one.