### Video Transcript

Given that π is the vector negative nine π’ minus seven π£, where π’ and π£ are two perpendicular unit vectors, find negative one-half multiplied by vector π.

In this question, weβre given a vector π and this vector is given in terms of two perpendicular unit vectors, vector π’ and vector π£. We need to use this to determine negative one-half multiplied by vector π. To answer this question, letβs start by looking at the expression weβre asked to evaluate. Since negative one-half is a scalar and π is a vector, this is scalar multiplication of a vector. And we have a lot of different tools for dealing with scalar multiplication of vectors. For example, because weβre told π’ and π£ are perpendicular unit vectors, we could represent π in terms of its components and then use this to answer our question.

However, this is not necessary. We can just represent π in the terms of π’ and π£ weβre given in the question. Negative one-half π is equal to negative one-half multiplied by negative nine π’ minus seven π£. Now the only property we need to use is that scalar multiplication distributes over vector addition and subtraction. Thereβs a few different ways of writing this down. Weβll write this as negative one-half multiplied by negative nine π’ plus negative one-half multiplied by negative seven π’.

And it is worth pointing out we couldβve subtracted these two vectors and instead had a coefficient of seven in front of π£. Both of these methods will work and will give us the same answer. Now all we have to do is simplify the coefficients of π’ and π£ by multiplying these together. We have negative one-half multiplied by negative nine is nine over two and negative one-half times negative seven is positive seven over two. And this gives us our final answer. If π is equal to negative nine π’ minus seven π£, then negative one-half times vector π is going to be equal to nine over two π’ plus seven over two π£.