Video Transcript
Fill in the blank: vectors 𝐀
equals one, two and vector 𝐁 equals negative two, one are what.
The first thing we might consider
checking is if vectors 𝐀 and 𝐁 are parallel. We can recall that any two vectors
𝐀 and 𝐁 are parallel if we can write that vector 𝐀 is equal to 𝑘 times vector 𝐁
for any scalar 𝑘, where 𝑘 is not equal to zero. So, let’s check if this true for
the given vectors. Is there a value 𝑘 for which the
vector one, two is equal to 𝑘 times negative two, one? We can simplify the vector on the
right-hand side as negative two 𝑘, 𝑘. We can then evaluate the 𝑥- and
𝑦-components separately, so the 𝑥-components would give us one is equal to
negative two 𝑘. Dividing this equation by negative
two on both sides would give us that negative one-half is equal to 𝑘. So, 𝑘 is equal to negative
one-half.
Let’s have a look at the
𝑦-components. This gives us the equation two is
equal to 𝑘 or, alternatively, 𝑘 is equal to two. However, we have two different
values of 𝑘, which means that there is no value of 𝑘 for which vector 𝐀 is equal
to 𝑘 times vector 𝐁. This means that these two vectors
are not parallel, so let’s see what else they might be. Let’s check if they’re
perpendicular. We recall that if the dot product
of two vectors is equal to zero, then those vectors are perpendicular. For any two vectors 𝐮 is equal to
𝑥 one, 𝑦 one and 𝐯 is equal to 𝑥 two, 𝑦 two, then the dot product 𝐮𝐯 is equal
to the product 𝑥 one 𝑥 two plus the product 𝑦 one 𝑦 two.
To calculate the dot product of our
two vectors 𝐀 and 𝐁, then we work out one times negative two plus two times
one. And we can simplify this to
negative two plus two which is equal to zero. For any two vectors 𝐮 and 𝐯, they
are perpendicular if the dot product 𝐮 dot 𝐯 is equal to zero. We have established that the dot
product of vectors 𝐀 and 𝐁 here is equal to zero. This means that we can fill in the
blank. Vectors 𝐀 is equal to one, two and
𝐁 is equal to negative two, one are perpendicular.
