Orthogonal Projections and Complements

This video tutorial explains how to use the orthogonal projection formula to determine a vector projection onto a plane in R3. It begins by defining the subspace W with an orthogonal basis and provides an example using specific vectors. The tutorial includes a graphical representation of the vectors and their projections, followed by detailed calculations of the orthogonal projection of vector X onto W and its orthogonal complement. Finally, it demonstrates how to calculate the distance from vector X to W using the magnitude of the projection onto the orthogonal complement.

What is the purpose of using the orthogonal projection formula in this example?

• To find the inverse of a matrix

• To determine the projection of a vector onto a plane

• To calculate the cross product of two vectors

• To find the angle between two vectors

Which vectors form the orthogonal basis for the subspace W in this example?

• Vectors (1, 0, 0) and (0, 1, 0)

• Vectors (2, 3, -2) and (1, 1, 1)

• Vectors (1, 0, -1) and (1, 1, 1)

• Vectors (2, 3, -2) and (1, 0, -1)

How do we verify that vectors U1 and U2 are orthogonal?

• By checking if their sum is zero

• By checking if their dot product is zero

• By checking if their difference is zero

• By checking if their cross product is zero

What is the result of the orthogonal projection of vector X onto W?

• Vector (3, 1, -1)

• Vector (0, 0, 0)

• Vector (2, 3, -2)

• Vector (1, 0, -1)

How is the projection of vector X onto the orthogonal complement of W calculated?

• By adding vector X and vector Xw

• By subtracting vector Xw from vector X

• By multiplying vector X by vector Xw

• By dividing vector X by vector Xw

What is the vector result of the projection of X onto the orthogonal complement of W?

• Vector (0, 0, 0)

• Vector (1, 1, 1)

• Vector (-1, 2, -1)

• Vector (2, 3, -2)

What is the final step in finding the distance from vector X to W?

• Finding the magnitude of the projection of X onto the orthogonal complement of W

• Calculating the dot product of vector X and W

• Subtracting the magnitudes of vector X and W

• Adding the magnitudes of vector X and W

What is the magnitude of the projection of vector X onto the orthogonal complement of W?

• Approximately 4.000 units

• Approximately 3.162 units

• Approximately 1.732 units

• Approximately 2.4495 units

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What does the magnitude of the orange vector represent in the graphical representation?

• The angle between vector X and W

• The distance from vector X to W

• The sum of vector X and W

• The product of vector X and W

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What is the significance of the orthogonal complement in this context?

• It represents the inverse of W

• It is the set of all vectors parallel to W

• It is the set of all vectors perpendicular to W

• It represents the sum of all vectors in W