Representation of Vectors in 2D and 3D

In 2D

• Coordinate Form:
  A vector in two dimensions is typically written as v = ⟨x,y⟩ or as a column:

v=[x/y]

Here, x and y are the components along the horizontal (x-axis) and vertical (y-axis) directions.

Graphical Representation:
Think of a vector as an arrow in the plane. Its tail is at the origin (or another reference point), and its head is at the point (x,y). The arrow’s length (magnitude) and its direction give you a complete picture.

In 3D

• Coordinate Form:
  In three dimensions, a vector is expressed as v = ⟨x,y,z⟩ or:

v=[x/y/z]

Now you have an extra component, z, for the depth.

• Graphical Representation:
  Picture it as an arrow in space. Its tail is at the origin (or another designated point), and its head is at (x, y, z). The vector’s magnitude is the “distance” from the tail to the head, and its direction is determined by the angles it makes with each axis.

• Unit Vectors:
  In both 2D and 3D, we often use unit vectors to indicate direction:

• In 2D: i=⟨1,0⟩ and j=⟨0,1⟩
• In 3D: Add k=⟨0,0,1⟩

Any vector can be expressed as a linear combination of these unit vectors. For example, in 3D:

v=xi+yj+zk

2. Operations on Vectors

Addition and Subtraction

• Addition:
  To add two vectors, simply add their corresponding components.
  For u = ⟨u1,u2,u3⟩ and v = ⟨v1,v2,v3):

u+v=⟨u1+v1, u2+v2, u3+v3⟩

Geometrically, this is like “tip-to-tail” addition—place the tail of one at the head of the other, and the resulting vector runs from the start of the first to the end of the second.

• Subtraction:
  Subtraction is just adding a negative vector:

u−v=⟨u1−v1, u2−v2, u3−v3⟩

Think of it as finding the vector from the tip of v to the tip of u.

Scalar Multiplication

• Definition:
  Multiplying a vector by a scalar (a real number) scales its magnitude without changing its direction (unless the scalar is negative, which flips the direction).
  • • • • • •  
  • kv=⟨kx, ky, kz⟩
  •  
• Example:
  If v = ⟨2,−3,4⟩ and k=3, then:

• 3v=⟨6,−9,12⟩

3. Dot Product (Scalar Product)

Definition and Formula

• In 2D and 3D:
  The dot product of two vectors u and v is defined as:

u⋅v=u1v1+u2v2+u3v3

(In 2D, omit the u3u_3 and v3v_3 components.)

• Geometric Interpretation:
  The dot product is also given by:

u⋅v=∣u∣ ∣v∣cos⁡θ

where θ is the angle between the vectors. This relation is useful for finding the angle between vectors or determining if they are perpendicular (if the dot product is 0, they’re orthogonal).

Applications

• Finding the Angle Between Vectors:
  Rearrange the dot product formula to solve for θtheta:

Θ = cos ‾1 (u.v/ |u| |v|)

• Testing Orthogonality:
  If u⋅v=0, then the vectors are perpendicular.

4. Cross Product (Vector Product)

Definition and Formula (3D Only)

• Cross Product:
  For vectors u = ⟨u₁,u₂,u₃) and v = ⟨v1,v2,v3⟩, the cross product is:

u×v=⟨u2v3−u3v2, u3v1−u1v3, u1v2−u2v1⟩

• Geometric Interpretation:
  The cross product produces a vector that is perpendicular to both u and v, following the right-hand rule. Its magnitude is:

∣u×v∣=∣u∣ ∣v∣sin⁡θ

where θ is the angle between u and v.

Applications

• Finding a Perpendicular Vector:
  Use the cross product to determine a normal vector to a plane defined by two non-parallel vectors.
• Area of a Parallelogram:
  The magnitude of the cross product gives the area of the parallelogram formed by u and v.
• Volume of a Parallelepiped:
  Combining the cross product with a dot product, you can find volumes using the scalar triple product: V=∣u⋅(v×w)

5. Applications in Geometry

Vectors are not just abstract concepts—they’re the Swiss Army knife of geometry. Here’s how they come into play:

Representing Points and Directions

• Position Vectors:
  Any point in space can be represented by a vector from the origin. This makes it easy to describe lines, planes, and curves in space.

Finding Distances and Midpoints

• Distance Between Two Points:
  Given position vectors a and b, the distance is:

• ∣b−a∣=√(b1−a1)2+(b2−a2)2+(b3−a3)2

• Midpoint Formula:
  The midpoint vector is: a+b/2

Angles and Projections

• Angle Between Vectors:
  Use the dot product method as discussed earlier.
• Projection of One Vector onto Another:
  The projection of u onto v is: projvu=(u⋅v∣v∣2)v

• This is vital when resolving forces or finding component vectors in physics and engineering.

Defining Planes

• Equation of a Plane:
  A plane can be defined by a point P(to the base) 0 and a normal vector n: n⋅(r−P0)=0

where r is the position vector of any point on the plane.

Intersecting Lines and Planes

• Line-Plane Intersection:
  Use vector equations and projections to determine the point at which a line meets a plane.
• Angle Between a Line and a Plane:
  Compute the angle using the line’s direction vector and the plane’s normal vector.

Real-World Examples

• Engineering:
  Vectors are used to model forces and stresses in structures.
• Computer Graphics:
  In 3D modeling and game development, vectors define object positions, lighting, and camera directions.
• Navigation:
  Vectors help in calculating trajectories, directions, and distances in navigation systems.

Final Thoughts

Vectors are the backbone of many mathematical and physical applications. Once you’re comfortable with representing and manipulating vectors in 2D and 3D, you unlock a powerful toolset for solving complex geometric problems. Whether you’re calculating the angle between forces, finding the intersection of planes, or just determining the distance between two points, the concepts covered here are indispensable.

Keep practicing with problems, and soon you’ll be flexing those vector skills like a pro. Remember: in the world of geometry, vectors help you cut through the clutter and see the underlying structure of space.