SOLUTION: Consider the following. u = i + 3j, v = −3i + j (a) Find u · v (b) Find the angle between u and v to the nearest degree.

Algebra ->  Trigonometry-basics -> SOLUTION: Consider the following. u = i + 3j, v = −3i + j (a) Find u · v (b) Find the angle between u and v to the nearest degree.      Log On


   

Question 1206879: Consider the following.
u = i + 3j, v = −3i + j
(a) Find u · v
(b) Find the angle between u and v to the nearest degree.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
u+=+i+%2B+3j, v+=+-3i+%2B+j
(a) Find u+%2A+v

To find the dot product of u and v, use the formula
u+%2A+v=+%28a%5B1%5D+%2Aa%5B2%5D%29+%2B+%28b%5B1%5D+%2Ab%5B2%5D%29+
where a%5B1%5D, a%5B2%5D are the coefficients of i and b%5B1%5D,b%5B2%5D are the coefficients of j.

In this case,
a%5B1%5D=1, b%5B1%5D=3
a%5B2%5D=-3, b%5B2%5D=1

the dot product of u+and v is
u+%2Av+=+%281+%2A%28+-3%29%29+%2B+%283+%2A1%29+=-3%2B3=0


(b) Find the angle between u+and v to the nearest degree.
To find the angle between u+and v , we use the formula:

cos%28theta%29+=+%28u+%2Av%29%2F%28abs%28u%29%2Aabs%28v%29%29

where abs%28u%29 and abs%28v%29+are the magnitudes of u+and v , respectively
the magnitude of u is sqrt%281%5E2%2B3%5E2%29=sqrt%2810%29
the magnitude of v+is sqrt%28%28-3%29%5E2%2B1%5E2%29=sqrt%2810%29

cos%28theta%29+=+0%2F%28sqrt%2810%29%2Asqrt%2810%29%29
cos%28theta%29+=+0
theta+=+cos%5E-1%280%29
theta+=+90°