Use a graphing utility with matrix capabilities to find the following, where [tex]u =(-3,2,1,1), v =(0,2,-1,-2)$[/tex], and [tex]w =(1,-2,2,3)$[/tex].
(a) [tex]u +4 v =[/tex] $\square$
(b) [tex]w -2 u =[/tex] $\square$
(c) [tex]3 v +\frac{1}{2} u - w =[/tex] $\square$
Answer (1)
Calculate u + 4 v : Multiply vector v by 4 and add the result to vector u , resulting in ( − 3 , 10 , − 3 , − 7 ) .
Calculate w − 2 u : Multiply vector u by 2 and subtract the result from vector w , resulting in ( 7 , − 6 , 0 , 1 ) .
Calculate 3 v + 2 1 u − w : Multiply vector v by 3 and vector u by 2 1 , add the two resulting vectors, and subtract vector w , resulting in ( − 2.5 , 9 , − 4.5 , − 8.5 ) .
The final results are: u + 4 v = ( − 3 , 10 , − 3 , − 7 ) , w − 2 u = ( 7 , − 6 , 0 , 1 ) , and 3 v + 2 1 u − w = ( − 2.5 , 9 , − 4.5 , − 8.5 ) . u + 4 v = ( − 3 , 10 , − 3 , − 7 ) , w − 2 u = ( 7 , − 6 , 0 , 1 ) , 3 v + 2 1 u − w = ( − 2.5 , 9 , − 4.5 , − 8.5 )
Explanation
Problem Setup We are given three vectors: u = ( − 3 , 2 , 1 , 1 ) , v = ( 0 , 2 , − 1 , − 2 ) , and w = ( 1 , − 2 , 2 , 3 ) . Our task is to perform the following vector operations:
(a) Calculate u + 4 v (b) Calculate w − 2 u (c) Calculate 3 v + 2 1 u − w
Calculating u + 4v (a) To find u + 4 v , we first multiply the vector v by the scalar 4, and then add the resulting vector to u .
4 v = 4 ( 0 , 2 , − 1 , − 2 ) = ( 0 , 8 , − 4 , − 8 )
Now, we add this to u :
u + 4 v = ( − 3 , 2 , 1 , 1 ) + ( 0 , 8 , − 4 , − 8 ) = ( − 3 + 0 , 2 + 8 , 1 + ( − 4 ) , 1 + ( − 8 )) = ( − 3 , 10 , − 3 , − 7 )
Calculating w - 2u (b) To find w − 2 u , we first multiply the vector u by the scalar 2, and then subtract the resulting vector from w .
2 u = 2 ( − 3 , 2 , 1 , 1 ) = ( − 6 , 4 , 2 , 2 )
Now, we subtract this from w :
w − 2 u = ( 1 , − 2 , 2 , 3 ) − ( − 6 , 4 , 2 , 2 ) = ( 1 − ( − 6 ) , − 2 − 4 , 2 − 2 , 3 − 2 ) = ( 7 , − 6 , 0 , 1 )
Calculating 3v + (1/2)u - w (c) To find 3 v + 2 1 u − w , we first multiply the vector v by the scalar 3 and the vector u by the scalar 2 1 , then add the two resulting vectors, and finally subtract w from the sum.
3 v = 3 ( 0 , 2 , − 1 , − 2 ) = ( 0 , 6 , − 3 , − 6 )
2 1 u = 2 1 ( − 3 , 2 , 1 , 1 ) = ( − 2 3 , 1 , 2 1 , 2 1 )
Now, we add these two vectors:
3 v + 2 1 u = ( 0 , 6 , − 3 , − 6 ) + ( − 2 3 , 1 , 2 1 , 2 1 ) = ( − 2 3 , 7 , − 2 5 , − 2 11 )
Finally, we subtract w from the result:
3 v + 2 1 u − w = ( − 2 3 , 7 , − 2 5 , − 2 11 ) − ( 1 , − 2 , 2 , 3 ) = ( − 2 3 − 1 , 7 − ( − 2 ) , − 2 5 − 2 , − 2 11 − 3 ) = ( − 2 5 , 9 , − 2 9 , − 2 17 ) = ( − 2.5 , 9 , − 4.5 , − 8.5 )
Final Answer Therefore, the results of the vector operations are:
(a) u + 4 v = ( − 3 , 10 , − 3 , − 7 ) (b) w − 2 u = ( 7 , − 6 , 0 , 1 ) (c) 3 v + 2 1 u − w = ( − 2.5 , 9 , − 4.5 , − 8.5 )
Examples
Vector operations are fundamental in computer graphics for manipulating objects in 3D space. For example, when designing a video game, vectors can represent the position, velocity, and acceleration of characters or objects. Adding vectors can combine movements, scaling them can change speed or size, and subtracting them can calculate relative positions. These operations are essential for creating realistic and interactive experiences in games and simulations.
