The video tutorial explains the concept of standard unit vectors in R3 and explores whether the vector (0, 4, 5) is contained in the span of vectors (i + j) and k. It provides both algebraic and geometric justifications. Algebraically, it demonstrates that no scalars exist to express (0, 4, 5) as a linear combination of (i + j) and k, as shown by a contradiction in the row-reduced matrix. Geometrically, it illustrates that the vector (0, 4, 5) does not lie in the plane formed by (i + j) and k, confirming it is not in the span.

What are the standard unit vectors in R3?

• a, b, c

• x, y, z

• p, q, r

• i, j, k

What is the problem statement regarding the vector 0, 4, 5?

• Is it orthogonal to i, j, k?

• Is it a unit vector?

• Is it in the span of i, j, k?

• Is it in the span of i + j, k?

What does the span of a set of vectors represent?

• The difference between the vectors

• The product of the vectors

• The sum of the vectors

• The set of all possible linear combinations of the vectors

What must exist for the vector 0, 4, 5 to be in the span of i + j and k?

• Two vectors

• Two scalars

• Two matrices

• Two equations

Tags

CCSS.8.EE.C.8B

What is the result of the augmented matrix in reduced row echelon form?

• A solution exists

• No solution exists

• A unique solution exists

• Infinite solutions exist

Tags

CCSS.HSA.REI.C.9

What does the contradiction in the third row of the matrix indicate?

• The system has a solution

• The system has infinite solutions

• The system is inconsistent

• The system has no solution

Why can't the vector 0, 4, 5 be written as a linear combination of i + j and k?

• Because it is orthogonal to i + j and k

• Because it is a zero vector

• Because the first two components must be the same

• Because it is not a unit vector

What does the geometric justification involve?

• A numerical calculation

• A theoretical explanation

• A matrix representation

• A graph of the vectors

Tags

CCSS.HSN.VM.A.1

What color is used to represent the vector 0, 4, 5 in the geometric justification?

• Green

• Purple

• Red

• Blue

What does the blue vector's position relative to the plane indicate?

• It is perpendicular to the plane

• It is not in the span

• It is in the span

• It is parallel to the plane