### Video Transcript

Given that 𝐀 is equal to eight, negative three, one; 𝐁 is equal to 16, negative six, 𝑛; and 𝐀 and 𝐁 are two parallel vectors, find 𝐀 plus 𝐁.

We recall that two vectors 𝐮 and 𝐯 are parallel if 𝐯 is equal to 𝑘 multiplied by 𝐮, where 𝑘 is a scalar constant. In this question, we are given two vectors 𝐀 and 𝐁. And as these are parallel, 𝐁 is equal to 𝑘 multiplied by 𝐀. This means that 16, negative six, 𝑛 is equal to 𝑘 multiplied by eight, negative three, one. We can multiply any vector by a scalar by multiplying each of the components by that scalar. The right-hand side becomes eight 𝑘, negative three 𝑘, 𝑘.

We know that if two vectors are equal, their individual components must be equal. This means that 16 is equal to eight 𝑘. Negative six is equal to negative three 𝑘. And 𝑛 is equal to 𝑘. Dividing both sides of the first equation by eight, we have 𝑘 is equal to two. Dividing both sides of the second equation by negative three once again gives us 𝑘 is equal to two. Substituting this value of 𝑘 into the third equation gives us 𝑛 equals two.

We can now calculate vector 𝐀 plus vector 𝐁. We do this by adding the vectors eight, negative three, one and 16, negative six, two. When adding two vectors, we add their corresponding components. Eight plus 16 is 24, negative three plus negative six is negative nine, and one plus two is equal to three.

If vectors 𝐀 and 𝐁 are parallel, 𝐀 plus 𝐁 is equal to 24, negative nine, three.