### Video Transcript

If vector 𝐀 is equal to one,
negative two, two; vector 𝐁 is equal to two, 𝑚, 𝑛; vector 𝐂 is equal to 𝑚, 𝑛,
𝑚 plus 𝑛; and vector 𝐀 is parallel to vector 𝐁, find the magnitude of vector
𝐂.

We will begin by calculating the
values of 𝑚 and 𝑛 using the fact that vectors 𝐀 and 𝐁 are parallel. We recall 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. The vector two, 𝑚, 𝑛 is equal to
𝑘 multiplied by one, negative two, two.

We can multiply a vector by a
scalar by multiplying each of the components of the vector by that scalar. This means that the right-hand side
of our equation becomes 𝑘, negative two 𝑘, two 𝑘. If any two vectors are equal, their
corresponding components are equal. This means that two is equal to 𝑘,
𝑚 is equal to negative two 𝑘, and 𝑛 is equal to two 𝑘. Substituting 𝑘 equals two, we see
that 𝑚 is equal to negative four. 𝑛 is equal to two multiplied by
two, which is equal to positive four. This means that vector 𝐁 is equal
to two, negative four, four. As vector 𝐂 was equal to 𝑚, 𝑛,
𝑚 plus 𝑛, this is equal to negative four, four, zero.

We are asked to find the magnitude
of this vector. We can calculate the magnitude of
any vector by finding the square root of the sum of the squares of its individual
components. The magnitude of vector 𝐂 is equal
to the square root of negative four squared plus four squared plus zero squared. Negative four squared and four
squared are both equal to 16, and zero squared is zero. Our expression simplifies to the
square root of 32.

Using our laws of radicals or
surds, this can be rewritten as the square root of 16 multiplied by the square root
of two. And as the square root of 16 is
four, this is equal to four root two. If vectors 𝐀 and 𝐁 are parallel,
then the magnitude of vector 𝐂 is four root two.