Which of the following is true of
the vectors 𝚨 equal to negative three, seven, negative eight and 𝚩 negative six,
negative one, negative one. Is it (A) they are parallel, (B)
they are perpendicular, or (C) they are neither parallel nor perpendicular?
In order to answer this question,
we need to recall the properties of two vectors when they’re parallel or
perpendicular. Two vectors are parallel if vector
𝚨 is equal to 𝑘 multiplied by vector 𝚩, where 𝑘 is a scalar quantity not equal
to zero. This means that each of their
components must be multiplied by the same scalar.
In this question, the value of 𝑘
would have to satisfy the three equations. Negative three is equal to negative
six 𝑘, seven is equal to negative one 𝑘, and negative eight is equal to negative
one 𝑘. It is clear from the second and
third equation that this will not be correct, as negative one 𝑘 cannot be equal to
seven and negative eight. Dividing both sides of the bottom
equation by negative one gives us 𝑘 is equal to eight, whereas dividing both sides
of the second equation by negative one gives us 𝑘 is equal to negative seven. If we divide both sides of the top
equation by negative six, we get 𝑘 is equal to one-half.
As the value of 𝑘 is not the same
for all three equations, we can conclude that vector 𝚨 is not equal to a scalar
quantity 𝑘 multiplied by vector 𝚩. This means that the two vectors are
not parallel and option (A) is incorrect.
We know that two vectors are
perpendicular if their dot product is equal to zero. We calculate the dot product of two
vectors by firstly multiplying their corresponding components. We then find the sum of these three
values. Negative three multiplied by
negative six is equal to 18. Seven multiplied by negative one is
negative seven. And adding this is the same as
subtracting seven. Finally, negative eight multiplied
by negative one is equal to eight. The dot product of vector 𝚨 and
vector 𝚩 is equal to 18 minus seven plus eight. This is equal to 19.
As this is not equal to zero, the
dot product of vector 𝚨 and vector 𝚩 is not equal to zero. We can, therefore, conclude that
the two vectors are not perpendicular, so option (B) is also incorrect. The correct answer is, therefore,
option (C). Vector 𝚨 and vector 𝚩 are neither
parallel nor perpendicular.