Q. 18 - Calculus

Question

Let \(v=\langle w,x,y,z\rangle\).Describe the sets of points in \(R^4\) satisfying \(||v||=4\).

Step-by-Step Solution

Answer

The sets of points in \(R^4\) satisfying the equation \(||v||=4\) lie on the surface \(w^2+x^2+y^2+z^2=4\).

Step 1. Given Information

The vector \(v=\langle w,x,y,z\rangle\). 

Step 2. Find the set of points

The points which are unit \(4\) distance from the origin in the set \(R^4\) will satisfy the equation

\(||v||=4\).

Therefore, all the position vectors with terminal point is unit \(4\) distance from the origin are part of the set.

Thus, the points will satisfy the following equation:

\(\sqrt{\left(w-0\right)^2+\left(x-0\right)^2 +\left(y-0\right)^2 +\left(z-0\right)^2 }=4\)

\(w^2+x^2+y^2+z^2=4\)

Thus, the sets of points in \(R^4\) satisfying the equation \(||v||=4\) lie on the surface \(w^2+x^2+y^2+z^2=4\).