This video tutorial explains the formula for determining the angle between two vectors using the dot product and the law of cosines. It begins with a review of the formula, followed by a geometric interpretation of vector subtraction. The video then applies the law of cosines to a vector triangle and introduces the dot product. Finally, it combines equations to derive the angle formula, providing a comprehensive understanding of the mathematical proof.

What is the formula used to determine the angle between two vectors?

• cosine theta = u cross v / (magnitude of u * magnitude of v)

• cosine theta = u dotted with v / (magnitude of u * magnitude of v)

• cosine theta = u - v / (magnitude of u * magnitude of v)

• cosine theta = u + v / (magnitude of u * magnitude of v)

How do you geometrically interpret v minus u?

• As a vector with its initial point at the terminal point of u

• As a vector with its initial point at the initial point of v

• As a vector with its terminal point at the initial point of v

• As a vector with its terminal point at the terminal point of u

What is the purpose of applying the law of cosines in this context?

• To solve for the dot product of u and v

• To calculate the magnitude of vector v

• To find the length of vector u

• To determine the angle between vectors u and v

What does the dot product of a vector with itself represent?

• Zero

• The angle between the vector and itself

• The square of the magnitude of the vector

• The magnitude of the vector

What is the result of expanding the dot product expression for v minus u?

• v dotted with v minus u dotted with u

• v dotted with u minus u dotted with v

• v dotted with v minus 2 times u dotted with v plus u dotted with u

• v dotted with u plus u dotted with v

How do you simplify the equation derived from the law of cosines and dot product?

• By multiplying both sides by the magnitude of u

• By subtracting the magnitudes of u and v

• By adding the magnitudes of u and v

• By canceling out the common terms on both sides

What is the final formula for cosine of the angle between two vectors?

• u dotted with v divided by the sum of magnitudes of u and v

• u dotted with v divided by the product of magnitudes of u and v

• u cross v divided by the product of magnitudes of u and v

• u plus v divided by the product of magnitudes of u and v

What happens to the negative twos in the final step of the proof?

• They cancel out

• They are multiplied by both sides

• They are added to both sides

• They are divided by both sides

What is the significance of the angle c in the context of this proof?

• It is the angle between vector v and the y-axis

• It is the angle between vector u and the origin

• It is the angle between vector u and the x-axis

• It is the angle between the two vectors u and v

What is the role of the law of cosines in this proof?

• To find the dot product of u and v

• To establish a relationship between the sides of a triangle and the cosine of one of its angles

• To calculate the magnitude of vector u

• To determine the direction of vector v