http://www.ck12.org Chapter 7. Vectors
There are two standard unit vectors that make up all other vectors in the coordinate plane. They are−→i which is
the vector< 1 , 0 >and−→j which is the vector< 0 , 1 >. These two unit vectors are perpendicular to each other. A
linear combination of−→i and−→j will allow you to uniquely describe any other vector in the coordinate plane. For
instance the vector< 5 , 3 >is the same as 5−→i + 3 −→j.
Often vectors are initially described as an angle and a magnitude rather than in component form. Working with vec-
tors written as an angle and magnitude requires extremely precise geometric reasoning and excellent pictures. One
advantage of rewriting the vectors in component form is that much of this work is simplified.
Example A
A plane has a bearing of 60◦and is going 350 mph. Find the component form of the velocity of the airplane.
Solution: A bearing of 60◦is the same as a 30◦on the unit circle which corresponds to the point
(√
23 ,^12
)
. When
written as a vector<
√ 3
2 ,^12 >is a unit vector because it has magnitude 1. Now you just need to scale by a factor
of 350 and you get your answer of< 175 √ 3 , 175 >.
Example B
Consider the plane flying in Example A. If there is wind blowing with the bearing of 300◦at 45 mph, what is the
component form of the total velocity of the airplane?
Solution: A bearing of 300◦is the same as 150◦on the unit circle which corresponds to the point
(
−
√ 3
2 ,^12
)
. You
can now write and then scale the wind vector.
45 ·<−
√ 3
2 ,^12 >=<−^45
√ 3
2 ,^452 >
Since both the wind vector and the velocity vector of the airplane are written in component form, you can simply
sum them to find the component vector of the total velocity of the airplane.
< 175
√
3 , 175 >+<−^45
√ 3
2 ,^452 >=<^305
√ 3
2 ,^3952 >
Example C
Consider the plane and wind in Example A and Example B. Find the actual ground speed and direction of the plane
(as a bearing).
Solution: Since you already know the component vector of the total velocity of the airplane, you should remember
that these components represent anxdistance and aydistance and the question asks for the hypotenuse.