Vectors are an integral part of mathematics, physics, and engineering. This comprehensive guide focuses on the essentials of vector notation and fundamentals, providing a solid foundation for students.

Vector Notation in 2D and 3D

In two dimensions (2D), vectors are expressed as $\mathbf{v} = xi + yj$, where $x$ and $y$ are the components along the x and y axes. This notation is pivotal for representing vectors in a plane.

In three dimensions (3D), vectors extend to $\mathbf{v} = xi + yj + zk$, incorporating the z-axis. This addition allows for the representation of vectors in a three-dimensional space, crucial for fields like physics and engineering.

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Directed Line Segments

Vectors are represented as directed line segments. This representation is significant for several reasons:

• Direction: The line's orientation indicates the vector's direction. For example, a vector pointing upwards represents a different direction than one pointing downwards.
• Magnitude: The length of the line segment corresponds to the vector's magnitude. A longer line indicates a larger magnitude.
• Visualization: This representation helps in visualizing vectors in physical contexts, such as forces acting on an object.

Example

Calculate the magnitude of the vector $\mathbf{v} = 6i - 8j$.

Solution:

1. Understand the Problem: We need to find the magnitude (or length) of the vector $\mathbf{v}$. The vector is given in terms of its components along the x-axis $6i$ and y-axis $-8j$.

2. Apply the Formula: The magnitude of a vector in 2D is calculated using the Pythagorean theorem. The formula is $|\mathbf{v}| = \sqrt{x^2 + y^2}$, where $x$and $y$ are the components of the vector.

3. Substitute the Values: Substitute 6 for x and -8 for y in the formula. So, $|\mathbf{v}| = \sqrt{6^2 + (-8)^2}$.

4. Calculate: Calculate the square of the components: $6^2 = 36$ and $(-8)^2 = 64$. Then sum them: $36 + 64 = 100$.

5. Final Step: Take the square root of 100, which is 10. So, $|\mathbf{v}| = 10$.

6. Conclusion: The magnitude of the vector $\mathbf{v}$ is 10.

Position, Displacement, and AB Vectors

Position Vectors: These vectors define a point's location relative to the origin.

Displacement Vectors: These describe the change in position of an object. They are fundamental in physics for representing motion.

Vectors AB: Vectors like AB denote the vector from point A to point B. They are essential in geometry for describing the relative positions of points.

Example

Determine the vector $\mathbf{AB}$ if point A has a position vector $4i + 2j$ and point B has a position vector $-i - 3j$.

Solution:

1. Understand the Problem: We need to find the vector $\mathbf{AB}$ which represents the direction and magnitude from point A to B.

2. Vector Subtraction Formula: The vector $\mathbf{AB}$ is calculated as $\mathbf{B} - \mathbf{A}$.

3. Substitute the Values: Substitute the position vectors of A and B into the formula: $\mathbf{AB} = (-i - 3j) - (4i + 2j)$.

4. Simplify: Perform the subtraction: $-i - 3j - 4i - 2j$.

5. Combine Like Terms: Combine the i and j components: $-5i - 5j$.

6. Conclusion: The vector $\mathbf{AB}$ is $-5i - 5j$.

Geometric Representation

Vectors are depicted geometrically, illustrating their role in translating, rotating, and scaling objects in space. This geometric representation is vital for understanding how vectors interact with physical objects and spaces.

• Translation: Vectors describe the movement of an object from one location to another. For example, a vector $3i + 4j$ can represent moving an object 3 units to the right and 4 units up.
• Rotation: Vectors can represent the rotation of objects around a point or axis. This is particularly useful in mechanical engineering and robotics.
• Scaling: In graphics and physics, vectors are used to scale objects, indicating the direction and magnitude of the scaling.

Example

A vector in 3D space is given as $\mathbf{v} = 2i - 3j + 4k$. Describe its components and their significance.

Solution:

1. Understand the Problem: The vector $\mathbf{v}$ is given in 3D space with components along the x, y, and z axes.

2. Identify Components: The vector has components 2 along the x-axis $i$, -3 along the y-axis $j$, and 4 along the z-axis $k$.

3. Significance of Components:

• The x-component $2i$ means the vector extends 2 units in the positive x-direction.
• The y-component $-3j$ means the vector extends 3 units in the negative y-direction (opposite to the positive y-axis).
• The z-component $4k$ means the vector extends 4 units in the positive z-direction.

4. Real-world Interpretation: Such a vector could represent various physical quantities in 3D space, like force, velocity, or displacement.

5. Conclusion: The vector $\mathbf{v} = 2i - 3j + 4k$ extends 2 units along the x-axis, -3 units along the y-axis, and 4 units along the z-axis. This could represent a physical quantity like force acting in 3D space.