AMD Day 21: Dot Product Name: $\qquad$ Date: $\qquad$ Period: Instructions: Find Each Resultant Vector In The Component And Linear Combination Forms. Then Find Its Magnitude.1) Given Vectors: ${ U = (-1, -7) }$

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Date: March 10, 2024 Period: Mathematics

Instructions:

Find each resultant vector in the component and linear combination forms. Then find its magnitude.

Problem 1: Given Vectors

Vectors u and v

We are given two vectors:

  • u=(−1,−7)\mathbf{u} = (-1, -7)
  • v=(3,2)\mathbf{v} = (3, 2)

Finding the Resultant Vector in Component Form

To find the resultant vector in component form, we need to add the corresponding components of the two vectors.

u+v=(−1+3,−7+2)=(2,−5)\mathbf{u} + \mathbf{v} = (-1 + 3, -7 + 2) = (2, -5)

Finding the Resultant Vector in Linear Combination Form

The resultant vector in linear combination form is given by:

u+v=1u+1v\mathbf{u} + \mathbf{v} = 1\mathbf{u} + 1\mathbf{v}

where 11 is the scalar multiplier.

Finding the Magnitude of the Resultant Vector

The magnitude of the resultant vector is given by:

∥u+v∥=(2)2+(−5)2=4+25=29\|\mathbf{u} + \mathbf{v}\| = \sqrt{(2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}

Discussion

The dot product of two vectors is a way to multiply two vectors and get a scalar value. It is defined as the sum of the products of the corresponding components of the two vectors.

u⋅v=(u1,u2)⋅(v1,v2)=u1v1+u2v2\mathbf{u} \cdot \mathbf{v} = (u_1, u_2) \cdot (v_1, v_2) = u_1v_1 + u_2v_2

In this case, we have:

u⋅v=(−1)(3)+(−7)(2)=−3−14=−17\mathbf{u} \cdot \mathbf{v} = (-1)(3) + (-7)(2) = -3 - 14 = -17

The dot product of two vectors can be used to find the angle between them. The formula for finding the angle between two vectors is:

cos⁡θ=u⋅v∥u∥∥v∥\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}

where θ\theta is the angle between the two vectors.

Problem 2: Finding the Angle Between Two Vectors

Find the angle between the vectors u\mathbf{u} and v\mathbf{v}.

Solution

To find the angle between the vectors, we need to find the dot product of the two vectors and the magnitudes of the two vectors.

u⋅v=(−1)(3)+(−7)(2)=−3−14=−17\mathbf{u} \cdot \mathbf{v} = (-1)(3) + (-7)(2) = -3 - 14 = -17

∥u∥=(−1)2+(−7)2=1+49=50\|\mathbf{u}\| = \sqrt{(-1)^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50}

∥v∥=(3)2+(2)2=9+4=13\|\mathbf{v}\| = \sqrt{(3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13}

Now, we can plug these values into the formula for finding the angle between two vectors:

cos⁡θ=−175013=−17650\cos \theta = \frac{-17}{\sqrt{50} \sqrt{13}} = \frac{-17}{\sqrt{650}}

To find the angle, we need to take the inverse cosine of both sides:

θ=cos⁡−1(−17650)\theta = \cos^{-1} \left( \frac{-17}{\sqrt{650}} \right)

Conclusion

In this article, we have discussed the dot product of two vectors and how to find the resultant vector in component and linear combination forms. We have also discussed how to find the magnitude of the resultant vector and the angle between two vectors. The dot product is a powerful tool in mathematics and has many applications in physics, engineering, and other fields.

References

  • [1] "Vector Calculus" by Michael Corral
  • [2] "Linear Algebra and Its Applications" by Gilbert Strang

Future Work

In the future, we can explore more advanced topics in vector calculus, such as the cross product and the triple product. We can also discuss the applications of vector calculus in physics and engineering.

Code

import math

u = (-1, -7)
v = (3, 2)



dot_product = u[0]*v[0] + u[1]*v[1]



magnitude_u = math.sqrt(u[0]**2 + u[1]**2)
magnitude_v = math.sqrt(v[0]**2 + v[1]**2)



angle = math.acos(dot_product / (magnitude_u * magnitude_v))


print("The angle between the vectors is:", math.degrees(angle))

Date: March 10, 2024 Period: Mathematics

Q&A Session:

Q1: What is the dot product of two vectors?

A1: The dot product of two vectors is a way to multiply two vectors and get a scalar value. It is defined as the sum of the products of the corresponding components of the two vectors.

Q2: How do I find the resultant vector in component form?

A2: To find the resultant vector in component form, you need to add the corresponding components of the two vectors.

Q3: How do I find the resultant vector in linear combination form?

A3: The resultant vector in linear combination form is given by:

u+v=1u+1v\mathbf{u} + \mathbf{v} = 1\mathbf{u} + 1\mathbf{v}

where 11 is the scalar multiplier.

Q4: How do I find the magnitude of the resultant vector?

A4: The magnitude of the resultant vector is given by:

∥u+v∥=(2)2+(−5)2=4+25=29\|\mathbf{u} + \mathbf{v}\| = \sqrt{(2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}

Q5: What is the formula for finding the angle between two vectors?

A5: The formula for finding the angle between two vectors is:

cos⁡θ=u⋅v∥u∥∥v∥\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}

where θ\theta is the angle between the two vectors.

Q6: How do I find the dot product of two vectors?

A6: To find the dot product of two vectors, you need to multiply the corresponding components of the two vectors and add them together.

Q7: What is the significance of the dot product in mathematics?

A7: The dot product is a powerful tool in mathematics and has many applications in physics, engineering, and other fields. It can be used to find the angle between two vectors, the magnitude of a vector, and the projection of one vector onto another.

Q8: Can you provide an example of how to use the dot product in a real-world scenario?

A8: Yes, the dot product can be used to find the angle between two vectors in a real-world scenario. For example, in physics, the dot product can be used to find the angle between the force applied to an object and the direction of motion of the object.

Q9: How do I use the dot product to find the angle between two vectors?

A9: To use the dot product to find the angle between two vectors, you need to follow these steps:

  1. Find the dot product of the two vectors.
  2. Find the magnitudes of the two vectors.
  3. Plug the values into the formula for finding the angle between two vectors.

Q10: Can you provide a code example of how to use the dot product in Python?

A10: Yes, here is a code example of how to use the dot product in Python:

import math


u = (-1, -7)
v = (3, 2)



dot_product = u[0]*v[0] + u[1]*v[1]



magnitude_u = math.sqrt(u[0]**2 + u[1]**2)
magnitude_v = math.sqrt(v[0]**2 + v[1]**2)



angle = math.acos(dot_product / (magnitude_u * magnitude_v))


print("The angle between the vectors is:", math.degrees(angle))

This code calculates the dot product, magnitudes, and angle between the vectors using the formulas discussed in this article.