Unit Vectors and Position Vectors (Introductory)

Introduction

Key Concepts

1. Understanding Vectors

A vector is a mathematical entity that has both magnitude and direction. Unlike scalars, which only possess magnitude, vectors are essential in representing quantities that involve direction, such as velocity, force, and displacement.

2. Position Vectors

A position vector specifies the position of a point in space relative to an origin. In a two-dimensional plane, the position vector of a point \( P(x, y) \) is represented as: $$ \vec{r} = x\mathbf{i} + y\mathbf{j} $$ Here, \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors along the x-axis and y-axis, respectively.

In three dimensions, the position vector for a point \( P(x, y, z) \) extends to: $$ \vec{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} $$ where \( \mathbf{k} \) is the unit vector along the z-axis.

3. Unit Vectors

A unit vector is a vector with a magnitude of one unit. It is used to specify direction without concerning magnitude. The standard unit vectors in three-dimensional space are:

• \( \mathbf{i} \): Unit vector along the x-axis.
• \( \mathbf{j} \): Unit vector along the y-axis.
• \( \mathbf{k} \): Unit vector along the z-axis.

Any vector can be expressed as a combination of these unit vectors. For example, a vector \( \vec{A} \) with components \( A_x \), \( A_y \), and \( A_z \) is written as: $$ \vec{A} = A_x\mathbf{i} + A_y\mathbf{j} + A_z\mathbf{k} $$

4. Magnitude of Vectors

The magnitude (length) of a vector \( \vec{A} \) is calculated using the Pythagorean theorem. In two dimensions: $$ |\vec{A}| = \sqrt{A_x^2 + A_y^2} $$ In three dimensions: $$ |\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} $$

5. Direction of Vectors

The direction of a vector is determined by the angle it makes with a reference axis. For instance, in two dimensions, the angle \( \theta \) between vector \( \vec{A} \) and the x-axis can be found using: $$ \theta = \tan^{-1}\left(\frac{A_y}{A_x}\right) $$ Understanding the direction is essential for resolving vectors into components and for vector addition.

6. Vector Addition and Subtraction

Vectors can be added or subtracted using their components. For two vectors \( \vec{A} = A_x\mathbf{i} + A_y\mathbf{j} \) and \( \vec{B} = B_x\mathbf{i} + B_y\mathbf{j} \):

• Addition: \( \vec{A} + \vec{B} = (A_x + B_x)\mathbf{i} + (A_y + B_y)\mathbf{j} \)
• Subtraction: \( \vec{A} - \vec{B} = (A_x - B_x)\mathbf{i} + (A_y - B_y)\mathbf{j} \)

Graphically, vector addition can be visualized using the "tip-to-tail" method, where the tail of the second vector is placed at the tip of the first vector.

7. Scalar Multiplication

A vector can be multiplied by a scalar (a real number), which scales the magnitude of the vector without altering its direction (unless the scalar is negative, which reverses the direction). $$ k\vec{A} = kA_x\mathbf{i} + kA_y\mathbf{j} + kA_z\mathbf{k} $$ Where \( k \) is a scalar.

8. Dot Product

The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is a scalar quantity defined as: $$ \vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z $$ It can also be expressed in terms of magnitudes and the angle between them: $$ \vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}| \cos\theta $$ The dot product is useful in finding angles between vectors and in projecting one vector onto another.

9. Cross Product

The cross product of two vectors \( \vec{A} \) and \( \vec{B} \) results in a third vector \( \vec{C} \) that is perpendicular to both \( \vec{A} \) and \( \vec{B} \). It is defined as: $$ \vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\mathbf{i} + (A_zB_x - A_xB_z)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k} $$ The magnitude of the cross product is: $$ |\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}| \sin\theta $$ Where \( \theta \) is the angle between \( \vec{A} \) and \( \vec{B} \).

10. Applications of Unit and Position Vectors

Unit and position vectors are fundamental in various applications:

• Physics: Representing forces, velocities, and accelerations.
• Engineering: Designing structures and analyzing stresses.
• Computer Graphics: Modeling object orientations and movements.
• Robotics: Calculating positions and movements of robots.

Comparison Table

Summary and Key Takeaways