Vector Projection and Orthogonal Components

This video tutorial explains vector projection, focusing on projecting one vector onto another. It uses a light and shadow analogy to illustrate the concept and discusses vector components, including orthogonal vectors. The video provides a formula for calculating vector projections and demonstrates examples in both 2D and 3D spaces. The tutorial concludes with a graphical representation of the concepts discussed.

What is the visual representation of projecting one vector onto another?

• A shadow cast by a light source

• A reflection in a mirror

• A rotation around an axis

• A translation along a plane

What is the vector component orthogonal to another vector called?

• Vector U

• Vector V

• Vector W sub 1

• Vector W sub 2

Which mathematical operation is used in the formula for vector projection?

• Dot product

• Matrix multiplication

• Vector addition

• Cross product

How is the magnitude of vector V squared represented in the formula for projection?

• As the sum of squares

• As the product of components

• As the difference of squares

• As the square root of the sum of squares

In vector projection, what does the dot product represent?

• A matrix value

• A vector value

• A complex number

• A scalar value

In the R2 example, what is the scalar multiple used to find the projection of U onto V?

• 3/5

• 14/10

• 7/10

• 1/2

What is the orthogonal component of vector U in the R2 example?

• (1/2, -1/2)

• (1/5, -2/5)

• (3/5, 2/5)

• (2/5, 1/5)

What is the result of adding the projection and orthogonal component of a vector?

• The original vector

• A zero vector

• A unit vector

• A perpendicular vector

In the R3 example, what is the projection of U onto V?

• (2/3, 1/3, 5/3)

• (1/3, 2/3, 3/3)

• (4/3, 5/3, 1/3)

• (5/3, 4/3, 2/3)

What is the orthogonal component of vector U in the R3 example?

• (2/3, 1/3, -2/3)

• (1/3, -2/3, 2/3)

• (1/3, 2/3, -1/3)

• (-2/3, 2/3, 1/3)