Understanding the Equation x = x2u + x3v: A Step-by-Step Analysis

The equation x = x2u + x3v with x2 and x3 free (and neither u nor v a multiple of the other), describes a plane through the origin.

To understand why the equation x = x2u + x3v describes a plane through the origin, let’s break it down and analyze it step by step

To understand why the equation x = x2u + x3v describes a plane through the origin, let’s break it down and analyze it step by step.

First, let’s understand the variables in the equation:

– x represents a point on the plane.
– x2 and x3 are coefficients that multiply with vectors u and v respectively.
– u and v are two independent direction vectors in the plane.

Now, let’s rewrite the equation with vectors:

x = x2u + x3v

Breaking it down further:

x = x2u means that x is a linear combination of vector u.
x = x3v means that x is a linear combination of vector v.

In other words, the equation states that any point x in the plane can be represented as a linear combination of vectors u and v. This means that the plane generated by x, u, and v will pass through the origin since the coefficient for the origin (0, 0) is zero.

To see this visually, imagine vectors u and v extending from the origin, and all points that lie on the plane can be reached by scaling and adding these vectors. Since the origin is at the starting point of u and v, any combination of these vectors would necessarily pass through the origin.

Additionally, x2 and x3 being free variables means that they can take any value, representing different linear combinations of vectors u and v that create the same plane through the origin.

To summarize, the equation x = x2u + x3v describes a plane through the origin because it represents all possible linear combinations of vectors u and v. The coefficients x2 and x3 allow for different combinations, and the fact that the equation is satisfied for any values of x2 and x3 ensures that the resulting plane passes through the origin.

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