Video Transcript
If 𝚨 equals negative five 𝐢 plus
𝑘𝐣, 𝚩 equals 𝐢 plus three 𝐣, and 𝚨 is perpendicular to 𝚩, then 𝑘 equals
what.
Here, given these vectors 𝚨 and 𝚩
and knowing that they’re perpendicular to one another, we want to solve for 𝑘,
which is this 𝑦-component of vector 𝚨. The key to solving for 𝑘 is
recognizing that since 𝚨 is perpendicular to 𝚩, then we know that the dot product
of these two vectors equals zero. This is always true for two vectors
that are perpendicular. And so in the case of vectors 𝚨
and 𝚩 whose components we know, we can write that negative five 𝑘 dotted with one,
three equals zero.
We’ll then begin to carry out this
dot product, starting by multiplying the corresponding components of the
vectors. Negative five times one plus 𝑘
times three equals zero, or in other words negative five plus three 𝑘 equals
zero. If we add five to both sides of
this equation, we get this result. And then dividing both sides by
three, we find that 𝑘 equals five-thirds. That’s our answer for 𝑘. So we can say that if 𝚨 equals
negative five 𝐢 plus 𝑘𝐣, 𝚩 equals 𝐢 plus three 𝐣, and 𝚨 is perpendicular to
𝚩, then 𝑘 equals five-thirds.
