If 𝚨 equals 12, eight; 𝚩 equals
three, 𝑚; 𝐂 equals 𝑛, nine; and 𝚨 is parallel to 𝚩, 𝚩 is perpendicular to 𝐂,
then 𝑚 plus 𝑛 equals what.
So, here we have these three
vectors 𝚨, 𝚩, and 𝐂, and we’re told that 𝚨 and 𝚩 are parallel while 𝚩 and 𝐂
are perpendicular. As a side note, this means that 𝚨
is also perpendicular to 𝐂. But we won’t need to use that fact
to solve for 𝑚 and 𝑛, these components of vectors 𝚩 and 𝐂, respectively. We can start out by solving for the
value of 𝑚 in vector 𝚩. Now, the fact that this vector is
parallel to vector 𝚨 means that we can write, for some nonzero constant 𝐶, that 𝐶
times 𝚩 equals 𝚨. Another way of saying this is that
the 𝑥-component of 𝐴 over the 𝑥-component of 𝐵 equals that same ratio for the
corresponding 𝑦-components of these vectors.
From the given information, we see
that 𝐴 sub 𝑥 equals 12, 𝐵 sub 𝑥 equals three, 𝐴 sub 𝑦 equals eight, while 𝐵
sub 𝑦 is 𝑚. So, 12 over three equals eight over
𝑚. And if we multiply both sides of
this equation by three and by 𝑚, we find that 12 times 𝑚 equals three times eight
or 24. Dividing both sides of this
equation by 12, we find that 𝑚 equals two. So, we now know part of our answer
of what 𝑚 plus 𝑛 will be. We can now use this information and
the fact that 𝚩 is perpendicular to vector 𝐂 to solve for 𝑛.
In general, if two vectors, we’ll
call them 𝐕 one and 𝐕 two, are perpendicular to one another, that means their dot
product is zero. So, since 𝚩 is perpendicular to
𝐂, we can write that 𝚩 dot 𝐂 equals zero, and therefore three, 𝑚 dot 𝑛, nine
equals zero. Before we calculate this dot
product, let’s substitute in our known value for 𝑚. We know 𝑚 is two, so three, two
dot 𝑛, nine equals zero. And now, carrying out this dot
product by first multiplying corresponding components of these vectors, we find that
three times 𝑛 plus two times nine is zero so that subtracting 18 from both sides,
we have that three times 𝑛 equals negative 18. Dividing both sides by three, we
find that 𝑛 equals negative six. We have then our values of 𝑚 and
𝑛, and adding them together, we get a result of negative four.
So, if 𝚨 equals 12, eight; 𝚩
equals three, 𝑚; 𝐂 equals 𝑛, nine; and 𝚨 is parallel to 𝚩, 𝚩 is perpendicular
to 𝐂, then 𝑚 plus 𝑛 equals negative four.