Prentice Hall Geometry

Chapter 8 Right Triangles and Trigonometry > 8-5 Vectors

You can find the magnitude and direction of a vector when the vector is described as an ordered pair.

Problem 3 Finding the Magnitude and Direction of a Vector

Multiple Choice An airplane lands 40 km west and 25 km south from where it took off. This trip can be described by the vector 〈 − 40 , − 25 〉 . left pointing angle bracket negative 40 comma negative 25 right pointing angle bracket . What are the approximate magnitude and direction of its flight vector?

A graph of a vector measuring c extends between the west and south axes, x degrees from the west axis, to a point negative 40 left of the origin and negative 25 below the origin.

47 km at 32 degrees south of west
47 km at 58 degrees south of west
2225 km at 32 degrees west of south
2225 km at 58 degrees west of south

Think	Write
The magnitude is the distance from the initial point to the terminal point. Use the Distance Formula to find the distance between (0, 0) and ( − 40 , − 25 ) . open negative 40 comma negative 25 close .	d = ( − 40 − 0 ) 2 + ( − 25 − 0 ) 2 = 1600 + 625 = 2225 ≈ 47.16990566 table with 4 rows and 2 columns , row1 column 1 , d , column 2 equals . square root of open . negative 40 minus 0 . close squared . plus . open . negative 25 minus 0 . close squared end root , row2 column 1 , , column 2 equals . square root of 1600 , plus 625 end root , row3 column 1 , , column 2 equals , square root of 2225 , row4 column 1 , , column 2 almost equal to . 47.16990566 , end table
The vector is x degrees south of west. Use the tangent ratio to find the angle formed by this vector.	tan x ° = 25 40 tangent x degrees equals , 25 over 40
Use a calculator to find the inverse tangent.	x = tan − 1 ( 25 40 ) ≈ 32.00538321 table with 2 rows and 2 columns , row1 column 1 , x , column 2 equals . tangent super negative 1 end super . open , 25 over 40 , close , row2 column 1 , , column 2 almost equal to . 32.00538321 , end table

The airplane flew about 47 km at 32 degrees south of west. The correct answer is A.

✓ Got It?

1. An airplane lands 246 mi east and 76 mi north from where it took off. What are the approximate magnitude and direction of its flight vector?
2. Reasoning Can a vector have a negative magnitude? Explain.

You can use a single lowercase letter, such as modified u with right harpoon up above , comma to name a vector.

The map below shows vectors representing a flight from Albuquerque to Salt Lake City, with a stopover in Flagstaff. The vector from Albuquerque to Salt Lake City is the sum, or resultant, of the other two vectors. You write this as w ⇀ = u ⇀ + v ⇀ . modified w with right harpoon up above , equals , modified u with right harpoon up above , plus , modified v with right harpoon up above , .

A map has vector u extending left from Albuquerque to Flagstaff, vector v extending up from Flagstaff to Salt Lake City, and vector w extending up to the left from Albuquerque to Salt Lake City.

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Problem 3 Finding the Magnitude and Direction of a Vector

✓ Got It?

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