Given that the vectors 𝐀 equals
three, 𝑥 plus one and 𝐁 equals negative two 𝑥, three are perpendicular, find the
value of 𝑥.
The key to this exercise here is to
recognize that if 𝐀 and 𝐁 are perpendicular, as they are, then that means that
their dot product is equal to zero. This is the case because any
vectors that are perpendicular don’t overlap one another at all. The dot product is essentially a
measure of overlap, so a dot product of zero between two vectors means that they are
In general, if we have two
two-dimensional vectors, we’ll call that 𝐕 one and 𝐕 two, then their dot product
is equal to the product of their 𝑥-components added to the product of their
𝑦-components. So when it comes to our given
vectors 𝐀 and 𝐁, their dot product equals the 𝑥-value of 𝐀 times the 𝑥-value of
𝐁 plus the 𝑦-value of 𝐀 times the 𝑦-value of 𝐁.
Multiplying out, we have negative
six 𝑥 plus three 𝑥 plus three or, simplifying, negative three 𝑥 plus three. We recall that this is equal to
zero since 𝐀 and 𝐁 are perpendicular. And so if we subtract three from
both sides of this equation, we find that negative three equals negative three
𝑥. And then we can divide both sides
by negative three to find that 𝑥 equals one. This is the value of 𝑥 for 𝐀 dot
𝐁 to be zero and, therefore, these vectors to be perpendicular.