### Video Transcript

Given that the vectors 𝐀 equal to
three, 𝑥 plus one and 𝐁 equal to negative two 𝑥, three are perpendicular, find
the value of 𝑥.

We begin by recalling that if two
vectors 𝐀 and 𝐁 are perpendicular, then their dot or scalar product is equal to
zero. We calculate this dot product by
finding the sum of the products of the corresponding components. In this question, we multiply three
by negative two 𝑥 and we multiply 𝑥 plus one by three. The sum of these two expressions is
the dot product of vectors 𝐀 and 𝐁. Multiplying three by negative two
𝑥 gives us negative six 𝑥. And multiplying 𝑥 plus one by
three gives us three 𝑥 plus three.

Collecting like terms, this
simplifies to negative three 𝑥 plus three. And since the two vectors are
perpendicular, we can set this expression equal to zero. We can then solve for 𝑥 by adding
three 𝑥 to both sides. Dividing through by three gives us
𝑥 is equal to one. If the vector 𝐀 equal to three, 𝑥
plus one is perpendicular to vector 𝐁 equal to negative two 𝑥, three, then the
value of 𝑥 is one.