For what value of 𝑘 are vectors 𝚨
seven, negative seven 𝑘, negative six and 𝚩 seven, negative three, 𝑘
We recall that the dot product of
any two vectors 𝚨 and 𝚩 is equal to the magnitude of vector 𝚨 multiplied by the
magnitude of vector 𝚩 multiplied by the cos of 𝜃, where 𝜃 is the angle between
the two vectors.
If two vectors are perpendicular,
as in this question, the angle between them will be equal to 90 degrees. We know that the cos of 90 degrees
is equal to zero. This leads us to the fact that if
two vectors are perpendicular, their dot product is equal to zero.
To find the dot products of two
vectors in three dimensions, we find the product of their corresponding components
and then the sum of these three values. Seven multiplied by seven is equal
to 49. Multiplying negative seven 𝑘 by
negative three gives us positive 21𝑘, as multiplying two negatives gives a positive
answer. Negative six multiplied by 𝑘 is
equal to negative six 𝑘. And adding this is the same as
subtracting six 𝑘.
This gives us the equation zero is
equal to 49 plus 21𝑘 minus six 𝑘. Subtracting 49 from both sides and
collecting like terms gives us negative 49 is equal to 15𝑘. Finally, dividing both sides of
this equation by 15 gives us 𝑘 is equal to negative 49 over 15. This is the value of 𝑘 such that
vectors 𝚨 and 𝚩 are perpendicular.