Fill in the blank: If the vectors 𝐀, which is equal to negative one, two, and 𝐁, which is equal to negative three, 𝑥, are parallel, then 𝑥 equals what.
We recall that if two vectors 𝐮 and 𝐯 are parallel, then vector 𝐮 is equal to 𝑘 multiplied by vector 𝐯, where 𝑘 is a scalar. In this question, we are given the two vectors negative one, two and negative three, 𝑥. As these are parallel, we will let vector 𝐁 equal 𝑘 multiplied by vector 𝐀. Negative three, 𝑥 is equal to 𝑘 multiplied by negative one, two.
We know that we can multiply a vector by a scalar by multiplying each of the components by that scalar. The right-hand side of our equation becomes negative 𝑘, two 𝑘. As this is equal to the vector negative three, 𝑥, the corresponding components must be equal. Considering the 𝑥-components, we have negative three is equal to negative 𝑘. Multiplying both sides of this equation by negative one, 𝑘 is equal to three. As the 𝑦-components must be equal, 𝑥 is equal to two 𝑘. As 𝑘 is equal to three, we have 𝑥 is equal to two multiplied by three, which in turn equals six. If the vectors 𝐀 and 𝐁 are parallel, then 𝑥 equals six.