Understanding Unit Vectors and Linear Combinations

The video tutorial introduces unit vectors, explaining their properties and how they relate to the unit circle. It demonstrates how to convert a vector into a unit vector by dividing it by its magnitude, using examples to illustrate the process. The tutorial also covers standard unit vectors and linear combinations, showing how vectors can be represented both graphically and algebraically.

What is the length of a unit vector?

• 2

• 3

• 0

• 1

Which of the following represents the unit vector i on the unit circle?

• (0, -1)

• (-1, 0)

• (1, 0)

• (0, 1)

How do you convert a vector with a magnitude of 10 into a unit vector?

• Multiply by 10

• Subtract 10

• Add 10

• Divide by 10

What is the magnitude of the vector (-2, 5) after converting it to a unit vector?

• √29

• 5

• 1

• 10

In the example of vector (6, -1), what is the magnitude before converting it to a unit vector?

• √36

• √37

• √39

• √38

What is the purpose of rationalizing a vector when converting it to a unit vector?

• To simplify calculations

• To remove radicals from the denominator

• To increase the vector's length

• To change the vector's direction

What is a linear combination of vectors?

• A vector with a magnitude of zero

• A sum of scalar multiples of vectors

• A vector with no direction

• A vector with a magnitude of one

How can you express a vector in component form?

• As a single number

• As a sum of unit vectors

• As a pair of coordinates

• As a product of vectors

If a vector u has an initial point at (2, -5) and a terminal point at (-1, 3), what is its component form?

• (1, -2)

• (3, -8)

• (-1, 2)

• (-3, 8)

What is the linear combination form of a vector with components (-6, -3)?

• 6i + 3j

• -6i + 3j

• 6i - 3j

• -6i - 3j