Understanding the Angle Between Two Vectors

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