Video Transcript
Which of the following straight
lines has a direction vector four, five, one and passes through the point five,
three, two? Option (A) 𝐫 equals five, three,
two plus 𝑡 times four, five, one. Option (B) 𝐫 equals negative five,
negative three, negative two plus 𝑡 times four, five, one. Is it option (C) 𝐫 equals negative
five, negative three, negative two plus 𝑡 times negative four, negative five,
one? Option (D) 𝐫 equals five, three,
two plus 𝑡 times negative four, negative five, one. Or is it option (E) 𝐫 equals four,
five, one plus 𝑡 times five, three, two?
In this question, we are given five
vector equations of lines and asked to determine which of these lines has a
direction vector four, five, one and passes through the point five, three, two.
To answer this question, we can
start by recalling that a direction vector of the line is any nonzero vector
parallel to the line. We can also recall that a vector
equation of a line is an equation of the form 𝐫 equals 𝐫 sub zero plus 𝑡 times
𝐝, where 𝐫 sub zero is the position vector of any point on the line and 𝐝 is any
direction vector of the line.
Vector equations of a line are not
unique, since we can choose any point on the line for the position vector 𝐫 sub
zero and any direction vector of the line for 𝐝. For example, we could have 𝐝 as
the vector four, five, one and 𝐫 sub zero as the vector five, three, two. This gives us that 𝐫 is equal to
the vector five, three, two plus 𝑡 times the vector four, five, one, which we can
see matches option (A).
For due diligence, we could check
the remaining options to see if they are possible vector equations of the line. If we did this, we would find that
options (C), (D), and (E) cannot be vector equations of the line since their
direction vectors are not parallel to the given direction vector.
Showing that option (B) is not a
possible vector equation of the line is more tricky. We would need to show that the
point with coordinates negative five, negative three, negative two does not lie on
the given line. We could do this by solving a
vector equation; however, it is not necessary. Instead, we can conclude that the
vector equation in option (A) has a direction vector of four, five, one and passes
through the point five, three, two, so it is a vector equation of this straight
line.
