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What is basis vector?
Basis vectors are fundamental vectors used to define a coordinate system or a vector space. They serve as the building blocks for representing other vectors within that space. Basis vectors are typically linearly independent, meaning that no basis vector can be expressed as a linear combination of the others.
In a cartesian coordinate system, The basis vectors are i-hat, j-hat and k-hat.
i-hat: This is the unit vector along the x-axis. which looks horizontally to the right in 2D and the positive x in 3D space.
j-hat: This is the unit vector along the y-axis
k-hat: This is the unit vector along the z-axis
Using this basis vectors we can represent any vector in the space. For example, if we want to represent the A vector in the space using this basis vectors we can do it as follows.
$$\vec{A}=x\hat{i}+y\hat{j}+z\hat{k}$$
where x,y, and z are the scaler components of vector A along the axis.
Considering there are two vectors b1 and b2. For b2 to be linearly independent of b1. b1 ≠ cb2 where c is a scaler component by which we can scale the value of b2.
Similarly, if there are 3 vectors b1,b2, and b3 for them to be linearly independent it should satisfy this rule.
$$b_1 + b_2 \neq b_3$$
we represent vectors in the cartesian plane on the basis of i-hat, j-hat, and k-hat but we can change the basis and represent vectors on the basis of any other vector.