Given that vector 𝐀 is equal to
negative two, three and vector 𝐁 is equal to negative one, eight, find the dot
product of 𝐀 minus 𝐁 and 𝐁.
We will begin this question by
subtracting vector 𝐁 from vector 𝐀. When subtracting vectors, we simply
need to subtract their individual corresponding components. Subtracting negative one from
negative two gives us negative one. This is because this is the same as
adding one to negative two. Subtracting eight from three gives
us negative five. The vector 𝐀 minus 𝐁 is therefore
equal to negative one, negative five.
We recall that in order to find the
dot or scalar product of two vectors, we begin by finding the product of their
individual components and then find the sum of these values. In this question, we need to find
the dot product of negative one, negative five and negative one, eight.
Both of the 𝑥-components of these
vectors are negative one. The 𝑦-components are negative five
and eight. Multiplying two negative numbers
gives us a positive answer, whereas multiplying a negative number by a positive
number gives a negative answer. This leaves us with one minus
40. Taking 40 away from one gives us an
answer of negative 39. This is the dot product of 𝐀 minus
𝐁 and 𝐁.