Question 1

Find   for the standard inner product defined in R3, where   .

Accepted Answer

A)       
B)       
C)      
D)       
E)       
A)       
B)       
C)      
D)       
E)

Question 2

Find the cross product   of the vectors   ,   .

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 3

Find the coordinates of   relative to the orthonormal basis   in R2.

Accepted Answer

A)     
B)    
C)    
D)     
E)     
A)     
B)    
C)    
D)     
E)

Question 4

Find the angle  in radians between the vectors   . Round your answers to three decimal places.

Accepted Answer

A)     
B)     
C)    
D)     
E)     
A)     
B)     
C)    
D)     
E)

Question 5

Find the vector v with length 2 and the same direction as the vector    .

Accepted Answer

A)    
B)    
C)    
D)    
E)     
A)    
B)    
C)    
D)    
E)

Question 6

Find the projection of the vector   onto the subspace

Accepted Answer

A)     
B)    
C)     
D)     
E)     
A)     
B)    
C)     
D)     
E)

Question 7

Use the Gram-Schmidt orthonormalization process to transform the basis   for R3 into an orthonormal basis. Use the Euclidean inner product for R3 and use the vectors in the order in which they are shown.

Accepted Answer

A)    
B)     
C)     
D)   
E)     
A)    
B)     
C)     
D)   
E)

Question 8

Find   , where u=   and v=   .

Accepted Answer

A)     
B)    
C)     
D)    
E)     
A)     
B)    
C)     
D)    
E)

Question 9

Determine whether the set of vectors   in R2 is orthogonal (but not orthonormal), orthonormal, or neither.

Accepted Answer

A)  orthonormal 
B)  neither 
C)  orthogonal 
A)  orthonormal 
B)  neither 
C)  orthogonal

Question 10

Find the quadratic least squares approximating function g for the function   . Round numerical values in your answer to four decimal places.

Accepted Answer

A)    
B)     
C)     
D)     
E)     
A)    
B)     
C)     
D)     
E)

Question 11

Find the angle  in radians between the vectors   for the inner product   . Round your answers to two decimal places.

Accepted Answer

A)    
B)    
C)     
D)    
E)     
A)    
B)    
C)     
D)    
E)

Question 12

Find   given   and   .

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 13

Find the area of the parallelogram that has the vectors   ,   as adjacent sides.

Accepted Answer

A)     
B)  400 
C)     
D)  4 
E)  800 
A)     
B)  400 
C)     
D)  4 
E)  800

Question 14

Use the inner product   to find   , where   and   .

Accepted Answer

A)     
B)     
C)     
D)    
E)     
A)     
B)     
C)     
D)    
E)

Find for the standard inner product defined in R³, where .

Find the cross product of the vectors , .

Find the coordinates of relative to the orthonormal basis in R².

Find the angle in radians between the vectors . Round your answers to three decimal places.

Find the vector v with length 2 and the same direction as the vector .

Find the projection of the vector onto the subspace

Use the Gram-Schmidt orthonormalization process to transform the basis for R³ into an orthonormal basis. Use the Euclidean inner product for R³ and use the vectors in the order in which they are shown.

Find , where u= and v= .

Determine whether the set of vectors in R² is orthogonal (but not orthonormal), orthonormal, or neither.

Find the quadratic least squares approximating function g for the function . Round numerical values in your answer to four decimal places.

Find the angle in radians between the vectors for the inner product . Round your answers to two decimal places.

Find given and .

Find the area of the parallelogram that has the vectors , as adjacent sides.

Use the inner product to find , where and .

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Exam 5: Inner Product Spaces

Find for the standard inner product defined in R3, where .