Video Transcript
The diagram shows a vector 𝐀 that
has a magnitude of 29. The angle between the vector and
the 𝑥-axis is 60 degrees. Give this vector in component
form. Round all of the numbers in your
answer to the nearest whole number.
In this question, we’re being asked
to give the vector 𝐀 in component form. This means that we need to find the
horizontal and vertical components of 𝐀, which we typically call 𝐴 subscript 𝑥
and 𝐴 subscript 𝑦. Then, we can write the vector 𝐀 in
the form 𝐀 is equal to 𝐴 subscript 𝑥 times 𝐢 hat plus 𝐴 subscript 𝑦 times 𝐣
hat. Here, 𝐢 hat and 𝐣 hat are the
unit vectors in the 𝑥- and 𝑦-directions.
Let’s first find the horizontal
component of the vector 𝐴 subscript 𝑥. To find this horizontal component,
recall that we can use the formula 𝐴 subscript 𝑥 is equal to 𝐴 times cos 𝜃,
where 𝐴 is the magnitude of the vector and 𝜃 is the argument of the vector. We are told in the question that
the magnitude of the vector 𝐀 is 29. We are also given an angle of 60
degrees between the vector and the negative 𝑥-axis. But we need to be careful here. This angle is not the argument of
the vector. The argument of a vector is defined
as the angle between the vector and the positive 𝑥-axis, measured counterclockwise
from the positive 𝑥-axis.
So, to find the argument of this
vector, we need to add 180 degrees to this angle of 60 degrees. Adding 60 degrees plus 180 degrees,
we find an argument 𝜃 of 240 degrees. Substituting these values for the
amplitude and argument into our equation, we find that 𝐴 subscript 𝑥 is equal to
29 multiplied by the cos of 240 degrees. Completing this calculation, we
find that the horizontal component 𝐴 subscript 𝑥 is equal to negative 14.5.
We want this answer to the nearest
whole number. So we can round down negative 14.5
to negative 15. We have then that the value of the
horizontal component of this vector is equal to negative 15. Let’s make a note of this value and
clear some space on screen.
We now need to find the vertical
component of the vector 𝐴 subscript 𝑦. To find this vertical component,
recall that we can use the formula 𝐴 subscript 𝑦 equals 𝐴 times sin 𝜃, where as
before 𝐴 is the magnitude and 𝜃 is the argument of the vector. The values of the magnitude and
argument of the vector are unchanged. So we can substitute those values
into this equation. And we find that the vertical
component 𝐴 subscript 𝑦 is equal to 29 multiplied by the sin of 240 degrees.
Completing this calculation, we
find that the vertical component 𝐴 subscript 𝑦 is equal to negative 25.11. Again, we want this to the nearest
whole number, so we can round up negative 25.11 to negative 25. This gives us the value of the
vertical component of the vector as negative 25.
We have now found both components
of this vector. And so the vector 𝐀 can be written
in component form as 𝐀 equals negative 15 times 𝐢 hat minus 25 times 𝐣 hat. This is our final answer to this
question.
