Assume {u1, U2, u3} does not span R³. Select the best statement. A. {U1, U2, u3, u4} never spans R³. B. {U1, U2, u3, u4} spans R³ unless u4 is the zero vector. C. {U1, U2, U3, U4} may, but does not have to, span R³. D. {U1, U2, U3, U4} spans R³ unless u is a scalar multiple of another vector in the set. ○ E. {U1, U2, u3, u4} always spans R³. F. none of the above

Let u be a linear combination of {1₁, 1₂, 13). Select the best statement. OA. {U₁, U₂, U3, U4} is a linearly dependent set of vectors unless one of {1₁, 12, 13} is the zero vector. OB. (U₁, U₂, U3, U4} is always a linearly dependent set of vectors. OC. {U₁, U₂, U3, U4} could be a linearly dependent or linearly dependent set of vectors depending on the vectors chosen. OD. {U₁, U₂, U3, U4} could be a linearly dependent or linearly dependent set of vectors depending on the vector space chosen. OE. {U₁, U₂, U3, U4} is never a linearly dependent set of vectors. OF. none of the above

Let u4 be a linear combination of {u₁, U2, U3}. Select the best statement. ● A. {u₁, U2, U3, U4} is a linearly dependent set of vectors unless one of {u₁, U2, U3} is the zero vector. ● B. {U1, U2, U3, U4} is never a linearly dependent set of vectors. ● C. {U1, U2, U3, u4} could be a linearly dependent or lin- early dependent set of vectors depending on the vectors chosen. ● D. {U1, U2, U3, u4} is always a linearly dependent set of vectors. ● E. {U1, U2, U3, U4} could be a linearly dependent or lin- early dependent set of vectors depending on the vector space chosen. ● F. none of the above

Let u and v be vectors in a Euclidean space. Select the best statement. A. The difference u - v may not be defined. B. The difference u – v is found by adding u to-v. C. The difference u - v cannot be computed by a simple formula. D. The difference u – v is found by adding -u to -v. E. The difference u – v is found by adding u to V. F. none of the above

Let X and Y be points in R³. Let P be a point on the line segment joining X and Y such that the distance XP is three times the distance YP. What is the position vector P in terms of the position vectors X and Y? (Use syntax like (-1/2)*X + (1/2)*Y .)

Assume {u1, U2, 13, 14} spans R³. Select the best statement. A. {U1, U2, u3} never spans R³ since it is a proper subset of a spanning set. B. {U1, U2, u3} spans R³ unless one of the vectors is the zero vector. C. {u1, U2, us} spans R³ unless one of the vectors is a scalar multiple of another vector in the set. D. {U1, U2, us} always spans R³. E. {U1, U2, u3} may, but does not have to, span R³. F. none of the above

) Let A = (4, -4, 4, 0) and B = (-4, 3, -4, 1) be points in R*. a. Find the vector AB in R“. help (vectors) b. Find a vector in the direction of AB that is 2 times as long as AB. help (vectors) c. Find a vector in the direction opposite AB that is 2 times as long as AB. help (vectors) d. Find a unit vector in the direction of AB. help (vectors) e. Find a vector in the direction of AB that has length 2. help (vectors)

Prove using vectors that the line joining one vertex of a parallelogram to the midpoint of an opposite side trisects a diagonal of the parallelogram.