Video Transcript
If a car moved 10 kilometers north
and then moved the same distance west, then the displacement of the car is
blank. Is it option (A) 10 kilometers
west, option (B) 10 kilometers northwest, option (C) 10 root two kilometers
northwest, or (D) 10 kilometers southwest?
One way of solving this problem is
using vectors. Let’s consider our coordinate axes
as shown, where the unit vector in the positive 𝑥-direction is equal to 𝐢 and the
unit vector in the positive 𝑦-direction is equal to 𝐣. We can then apply this information
to our four main compass points. Moving one kilometer in the
easterly direction will be equal to the unit vector 𝐢, and moving one kilometer in
a northerly direction will be equal to the unit vector 𝐣. This means that moving one
kilometer west will be equal to negative 𝐢 and moving one kilometer south will be
equal to negative 𝐣.
The car in this question begins by
moving 10 kilometers north. This is equal to the vector
10𝐣. The car then travels the same
distance west. This is equal to the vector
negative 10𝐢. Traveling 10 kilometers north is
equal to the vector 10𝐣. And traveling 10 kilometers west is
equal to the vector negative 10𝐢. We are asked to work out the
displacement 𝐬. This vector is equal to 10𝐣 plus
negative 10𝐢. This can be rewritten as negative
10𝐢 plus 10𝐣.
We can then find the magnitude of
this displacement vector by finding the sum of the squares of its components and
then square rooting our answer. The magnitude of the displacement
is equal to the square root of negative 10 squared plus 10 squared. This is equal to the square root of
200, which in turn simplifies to 10 root two. This is because the square root of
200 can be rewritten as the square root of 100 multiplied by the square root of
two. And we know the square root of 100
is 10.
As only one of our possible answers
has the value 10 root two, this suggests the correct answer is option (C). We know that the displacement of
the car is definitely 10 root two kilometers. From our initial diagram, we see
that we have a right isosceles triangle. This means that the angle or
direction the car is traveling is 45 degrees west of north. This is the compass point
northwest.
The correct answer is option
(C). If a car moves 10 kilometers north
and then the same distance west, its displacement is 10 root two kilometers
northwest.
