Vector Equations

Vector equations use direction vectors and variables to describe planes or lines in three-dimensional space. A vector with a magnitude of 1 is a unit vector. In a three-dimensional space, all positions are described using 3 axes, the x-axis, y-axis, and z-axis, and each axis has a unit vector. The unit vector for the x-axis is \(\hat {i}\), the unit vector for the y-axis is \(\hat {j}\), and the unit vector for the z-axis is \(\hat {k}\).


In a 3D space, a vector is written as: 
\(\vec{r} = {x {\hat {i}}} + {y {\hat {j}}} + {z {\hat {k}}}\)Where x, y, and z represent the scalar components of the vector.
The vector equation of a line in three-dimensional space is:  \(\vec{r} = \vec{a} + \lambda{{\vec{b}} }\) where \(λ∈R\)                                                                   
The vector equation of a plane in three-dimensional space is: \(\vec{r} \cdot {\vec n} = d\).       

A Cartesian equation is an algebraic expression that shows the relationship between variables (typically x, y, and z) using the Cartesian coordinate system. It is used to describe geometric figures such as lines, curves, and surfaces in terms of their positions on a coordinate plane or in space. Here are some key differences between vector and Cartesian equations.
 

In a three-dimensional space, the vector equation of a line uses vectors to describe all the points that lie on a straight line.
The vector equation of a line that passes through a single point:
                           \(\vec r = \vec a + \lambda \vec b \)
Here,
\(\vec r\) represents the position vector of any point on the line
While \(\vec a\) represents the position vector of a fixed point through which the line passes
\(\vec b\) = direction vector of the line
and \(\lambda\) = scalar parameter that varies.

Vector equation for a line passing through two points:
        \(\vec r  = \vec a + \lambda(\vec b - \vec a)\)

                            
Here, 
\(\vec a\) (position vector of point A)
\(\vec a = x_1 \hat i + y_1 \hat j + z_1 \hat k \)
\(\vec b\) (position vector of point B)                                          
  \(\vec b = x_2 \hat i + y_2 \hat j + z_2 \hat k \)

The direction vector is \(\vec b - \vec a\), pointing from point A to point B.
\(λ∈R\) is a scalar parameter
 

The vector equation represents motion from a fixed point in a plane along two independent directions.
  \(\vec r = \vec a + s \vec u + t \vec v\)

Here, \(\vec r\) is the position vector of any point P(x, y, z) lying on the plane
\(\vec a\) is vector of a specific point on the plane
\(\vec u, \vec v\) are two non-parallel direction vectors that lie on the plane.
s and t are scalar parameters; they can be any real number.
 

Vector equations help us represent lines, planes, and directions in space using vectors. In this section, we will discuss some tips and tricks to master vector equations. This help students to solve problems in physics and mathematics much more easily.

• Always separate the components, that is, break the vectors into x, y, and z components before solving. 
  
   

• use unit vectors, express the vectors as combinations of i, j, and k. It simplifies addition, subtraction, and scalar multiplication. 
  
   

• Memorize the basic formulas like: 
  Angle between vectors: \(cos{\theta} = {{{\vec a \cdot \vec b} \over  |\vec a| |\vec b|}}\). 
  Perpendicular vectors: Dot product = 0 \({(\vec a \cdot \vec b = 0 )}\)Magnitude: \({{|\vec v| = \sqrt {x^2 + y^2 + z^2}}}\)
  
   

• If a vector is a multiple of another vector, then the vectors are parallel. For example, (2, 4) is parallel to (1, 2).
  
   

• Students can draw a simple diagram to visualize how the vectors look, where they point, and how they relate to each other. This help students to understand if they are parallel, perpendicular, or intersecting.

Vector equations represent quantities having both magnitude and direction. They play a crucial role in various real-life, direction-based calculations:

• Force and motion analysis: Vector equations describe the position, velocity, and acceleration of moving objects in space. This helps analyze trajectories, forces, and dynamics in mechanics.
  
   
• Graphics and Animations: Vectors represent object positions, directions, and transformations in 2D or 3D space. They are used in calculations for moving graphic objects and rendering in games or simulations.

                         \(\vec p_{new} = \vec p _{original} + \vec v \cdot  t \)
 

• GPS and Navigation Systems: GPS uses vector math to compute displacement, direction, and distance between two points on Earth, determining shortest paths.

• Robotics: Vector equations are used in machinery like robotic arms to reach a point in 3D space. Movement and orientation are modeled using vector equations for precise positioning. 
  
   
• Engineering- structural analysis: Engineers use vectors to model forces acting on structures such as bridges and buildings. Vector equations help in calculating stress, strain, and stability.