✓ Got It?

2. 1. Rotate the vector v = 〈 − 3 , 5 〉 v equals left pointing angle bracket negative 3 comma 5 right pointing angle bracket  by 270 ° 270 degrees . What is the component form of the resulting vector?
   2. Reasoning What other matrix transformations can you apply to vectors in matrix form?

Take Note Properties: Operations With Vectors

Given v = 〈 v 1 , v 2 〉 , w = 〈 w 1 , w 2 〉 , v equals left pointing angle bracket , v sub 1 , comma , v sub 2 , right pointing angle bracket comma w equals left pointing angle bracket , w sub 1 , comma , w sub 2 , right pointing angle bracket comma  and any real number k:

Image Long Description

v + w = 〈 v 1 + w 1 , v 2 + w 2 〉 v − w = 〈 v 1 − w 1 , v 2 − w 2 〉 k v = 〈 k v 1 , k v 2 〉 table with 3 rows and 1 column , row1 column 1 , v plus w equals left pointing angle bracket , v sub 1 , plus , w sub 1 , comma , v sub 2 , plus , w sub 2 , right pointing angle bracket , row2 column 1 , v minus w equals left pointing angle bracket , v sub 1 , minus , w sub 1 , comma , v sub 2 , minus , w sub 2 , right pointing angle bracket , row3 column 1 , k v equals left pointing angle bracket k , v sub 1 , comma k , v sub 2 , right pointing angle bracket , end table

Problem 3 Adding and Subtracting Vectors

GRIDDED RESPONSE

Let u = 〈 − 2 , 3 〉 u equals left pointing angle bracket negative 2 comma 3 right pointing angle bracket  and v = 〈 5 , − 2 〉 . v equals left pointing angle bracket 5 comma negative 2 right pointing angle bracket .  What is | u + v u plus v |, rounded to the nearest hundredth?

To find u + v u plus v , use the tip-to-tail method shown above.

Image Long Description

• Step 1 Draw u = 〈 − 2 , 3 〉 u equals left pointing angle bracket negative 2 comma 3 right pointing angle bracket  in standard position.

  Think

  How do you draw v with initial point ( − 2 , 3 ) ? open negative 2 comma 3 close question mark

  Use ( − 2 + 5 , 3 − 2 ) = ( 3 , 1 ) open negative 2 plus 5 comma 3 minus 2 close equals open 3 comma 1 close  for the terminal point.

• Step 2 At the tip of u, draw v = 〈 5 , − 2 〉 v equals left pointing angle bracket 5 comma negative 2 right pointing angle bracket  from ( − 2 , 3 ) open negative 2 comma 3 close  to (3, 1).
• Step 3 Draw u + v u plus v to have the initial point of u and the terminal point of v.
• Step 4 Express u + v u plus v in component form. u + v u plus v = 〈3, 1〉
• Step 5 | u + v | = 3 2 + 1 2 = 10 ≈ 3.16 vertical line u plus v vertical line equals . square root of 3 squared , plus , 1 squared end root . equals square root of 10 almost equal to , 3.16

Check u + v = 〈 − 2 , 3 〉 + 〈 5 , − 2 〉 = 〈 − 2 + 5 , 3 + ( − 2 ) 〉 = 〈 3 , 1 〉 u plus v equals left pointing angle bracket negative 2 comma 3 right pointing angle bracket plus left pointing angle bracket 5 comma negative 2 right pointing angle bracket equals left pointing angle bracket negative 2 plus 5 comma 3 plus open negative 2 close right pointing angle bracket equals left pointing angle bracket 3 comma 1 right pointing angle bracket

✓ Got It?

3. Using the vectors given in Problem 3, what is | u − v | ? vertical line u minus v vertical line question mark