True or false: If vector 𝐀 is equal to two, one; vector 𝐁 is equal to negative six, 𝑥; and the vectors are parallel, then 𝑥 equals negative three.
We recall that two vectors 𝐮 and 𝐯 are parallel if 𝐮 is equal to 𝑘 multiplied by 𝐯, where 𝑘 is a scalar quantity. In this question, we are given two parallel vectors two, one and negative six, 𝑥. For these vectors to be parallel, vector 𝐁 must equal 𝑘 multiplied by vector 𝐀. Negative six, 𝑥 must therefore be equal to 𝑘 multiplied by two, one. In order to multiply a vector by a scalar, we multiply each of the components by that scalar. 𝑘 multiplied by the vector two, one is equal to the vector two 𝑘, 𝑘.
As this is equal to negative six, 𝑥, the corresponding components must be equal. This means that negative six must be equal to two 𝑘. Dividing both sides of this equation by two, we see that 𝑘 is equal to negative three. As 𝑥 is equal to 𝑘, 𝑥 must also be equal to negative three. We can therefore conclude that if vector 𝐀 is equal to two, one; vector 𝐁 is equal to negative six, 𝑥; and the two vectors are parallel, then 𝑥 does equal negative three. The statement, and therefore the correct answer, is true.