Given that 𝐀 is the vector negative nine 𝐢 minus seven 𝐣, where 𝐢 and 𝐣 are two perpendicular unit vectors, find negative one-half multiplied by vector 𝐀.
In this question, we’re given a vector 𝐀 and this vector is given in terms of two perpendicular unit vectors, vector 𝐢 and vector 𝐣. We need to use this to determine negative one-half multiplied by vector 𝐀. To answer this question, let’s start by looking at the expression we’re asked to evaluate. Since negative one-half is a scalar and 𝐀 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 we’re told 𝐢 and 𝐣 are perpendicular unit vectors, we could represent 𝐀 in terms of its components and then use this to answer our question.
However, this is not necessary. We can just represent 𝐀 in the terms of 𝐢 and 𝐣 we’re given in the question. Negative one-half 𝐀 is equal to negative one-half multiplied by negative nine 𝐢 minus seven 𝐣. Now the only property we need to use is that scalar multiplication distributes over vector addition and subtraction. There’s a few different ways of writing this down. We’ll write this as negative one-half multiplied by negative nine 𝐢 plus negative one-half multiplied by negative seven 𝐢.
And it is worth pointing out we could’ve subtracted these two vectors and instead had a coefficient of seven in front of 𝐣. Both of these methods will work and will give us the same answer. Now all we have to do is simplify the coefficients of 𝐢 and 𝐣 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 𝐀 is equal to negative nine 𝐢 minus seven 𝐣, then negative one-half times vector 𝐀 is going to be equal to nine over two 𝐢 plus seven over two 𝐣.