Work Out The Following Problems Accordingly:1. Let \[$ A = (2,3), B = (3,5) \$\], Then Find: A) The Position Vector Of \[$ AB \$\] And \[$ BA \$\]. B) The Direction And Magnitude Of \[$ AB \$\].2. Let \[$ U =

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Introduction

Vectors are mathematical objects that have both magnitude and direction. They are used to represent quantities with both size and direction, such as displacement, velocity, and acceleration. In this article, we will work out the problems related to vector operations and calculations.

Problem 1: Position Vectors of AB and BA

Let's consider two points A and B in a 2D plane, where A = (2,3) and B = (3,5). We need to find the position vectors of AB and BA.

Position Vector of AB

The position vector of AB is given by the difference between the coordinates of B and A.

ABβ†’=OBβ†’βˆ’OAβ†’\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}

where O is the origin.

ABβ†’=(3,5)βˆ’(2,3)\overrightarrow{AB} = (3,5) - (2,3)

AB→=(1,2)\overrightarrow{AB} = (1,2)

So, the position vector of AB is (1,2).

Position Vector of BA

The position vector of BA is given by the difference between the coordinates of A and B.

BAβ†’=OAβ†’βˆ’OBβ†’\overrightarrow{BA} = \overrightarrow{OA} - \overrightarrow{OB}

BAβ†’=(2,3)βˆ’(3,5)\overrightarrow{BA} = (2,3) - (3,5)

BAβ†’=(βˆ’1,βˆ’2)\overrightarrow{BA} = (-1,-2)

So, the position vector of BA is (-1,-2).

Problem 1: Direction and Magnitude of AB

Now, let's find the direction and magnitude of AB.

Direction of AB

The direction of AB is given by the unit vector in the direction of AB.

AB^=ABβ†’βˆ£ABβ†’βˆ£\hat{AB} = \frac{\overrightarrow{AB}}{|\overrightarrow{AB}|}

where ∣ABβ†’βˆ£|\overrightarrow{AB}| is the magnitude of AB.

AB^=(1,2)12+22\hat{AB} = \frac{(1,2)}{\sqrt{1^2 + 2^2}}

AB^=(1,2)5\hat{AB} = \frac{(1,2)}{\sqrt{5}}

So, the direction of AB is (1,2)5\frac{(1,2)}{\sqrt{5}}.

Magnitude of AB

The magnitude of AB is given by the length of the vector AB.

∣ABβ†’βˆ£=12+22|\overrightarrow{AB}| = \sqrt{1^2 + 2^2}

∣ABβ†’βˆ£=5|\overrightarrow{AB}| = \sqrt{5}

So, the magnitude of AB is 5\sqrt{5}.

Problem 2: Vector u

Let's consider a vector u = (a,b). We need to find the magnitude and direction of u.

Magnitude of u

The magnitude of u is given by the length of the vector u.

∣u∣=a2+b2|u| = \sqrt{a^2 + b^2}

So, the magnitude of u is a2+b2\sqrt{a^2 + b^2}.

Direction of u

The direction of u is given by the unit vector in the direction of u.

u^=u∣u∣\hat{u} = \frac{u}{|u|}

u^=(a,b)a2+b2\hat{u} = \frac{(a,b)}{\sqrt{a^2 + b^2}}

So, the direction of u is (a,b)a2+b2\frac{(a,b)}{\sqrt{a^2 + b^2}}.

Conclusion

In this article, we worked out the problems related to vector operations and calculations. We found the position vectors of AB and BA, and the direction and magnitude of AB. We also found the magnitude and direction of a vector u. These calculations are essential in physics and engineering to represent quantities with both size and direction.

References

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

Further Reading

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

Introduction

Vectors are mathematical objects that have both magnitude and direction. They are used to represent quantities with both size and direction, such as displacement, velocity, and acceleration. In this article, we will answer some frequently asked questions related to vector operations and calculations.

Q: What is a vector?

A: A vector is a mathematical object that has both magnitude and direction. It is used to represent quantities with both size and direction, such as displacement, velocity, and acceleration.

Q: How do you add two vectors?

A: To add two vectors, you need to add their corresponding components. For example, if you have two vectors u = (a,b) and v = (c,d), then the sum of u and v is given by:

u + v = (a + c, b + d)

Q: How do you subtract two vectors?

A: To subtract two vectors, you need to subtract their corresponding components. For example, if you have two vectors u = (a,b) and v = (c,d), then the difference of u and v is given by:

u - v = (a - c, b - d)

Q: What is the magnitude of a vector?

A: The magnitude of a vector is the length of the vector. It is given by the square root of the sum of the squares of its components. For example, if you have a vector u = (a,b), then the magnitude of u is given by:

|u| = √(a^2 + b^2)

Q: What is the direction of a vector?

A: The direction of a vector is the direction in which the vector points. It is given by the unit vector in the direction of the vector. For example, if you have a vector u = (a,b), then the direction of u is given by:

uΜ‚ = (a, b) / √(a^2 + b^2)

Q: How do you find the position vector of a point?

A: To find the position vector of a point, you need to subtract the coordinates of the origin from the coordinates of the point. For example, if you have a point P = (x,y), then the position vector of P is given by:

OP = (x, y) - (0, 0) = (x, y)

Q: How do you find the distance between two points?

A: To find the distance between two points, you need to find the magnitude of the vector between the two points. For example, if you have two points P = (x1, y1) and Q = (x2, y2), then the distance between P and Q is given by:

PQ = √((x2 - x1)^2 + (y2 - y1)^2)

Q: What is the dot product of two vectors?

A: The dot product of two vectors is a scalar value that represents the amount of "similarity" between the two vectors. It is given by the sum of the products of their corresponding components. For example, if you have two vectors u = (a,b) and v = (c,d), then the dot product of u and v is given by:

u Β· v = ac + bd

Q: What is the cross product of two vectors?

A: The cross product of two vectors is a vector value that represents the amount of "perpendicularity" between the two vectors. It is given by the determinant of a 2x2 matrix formed by the components of the two vectors. For example, if you have two vectors u = (a,b) and v = (c,d), then the cross product of u and v is given by:

u Γ— v = (ad - bc, ac - bd)

Conclusion

In this article, we answered some frequently asked questions related to vector operations and calculations. We hope that this article has helped you to understand the basics of vector operations and calculations.

References

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

Further Reading

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