Question

Consider a rectangle with vertices at E, F, G, and H. Suppose EF~ = p and FG = q. Express each of the vectors EH, GH~, FH~, and GE~ in terms of ~p and ~q.

Answer 1

`\[ EH = (p)/(√(p^2+q^2)) \cdot \mathbf{p} + (q)/(√(p^2+q^2)) \cdot \mathbf{q} \]`

`\[ GH~ = (p)/(√(p^2+q^2)) \cdot \mathbf{p} - (q)/(√(p^2+q^2)) \cdot \mathbf{q} \]`

`\[ FH~ = -(q)/(√(p^2+q^2)) \cdot \mathbf{p} + (p)/(√(p^2+q^2)) \cdot \mathbf{q} \]`

`\[ GE~ = -(q)/(√(p^2+q^2)) \cdot \mathbf{p} - (p)/(√(p^2+q^2)) \cdot \mathbf{q} \]`

Step-by-step explanation:

In a rectangle, opposite sides are parallel, and EF, GH, FH, and GE represent vectors in the rectangle. The vector EH is a linear combination of vectors \(\mathbf{p}\) and \(\mathbf{q}\), where the coefficients are determined by the proportions of the sides EF and FG. The magnitude of EH is given by the square root of the sum of the squares of the coefficients of \(\mathbf{p}\) and \(\mathbf{q}\).

For GH~, the vector is similar to EH but with the coefficient of \(\mathbf{q}\) negated, as GH is parallel and equal in magnitude to EF. Similarly, FH~ and GE~ have coefficients based on the proportions of the sides and are adjusted accordingly.

The expressions are normalized by dividing each term by the magnitude of EH to ensure that the resulting vectors have a magnitude of 1. This normalization makes the expressions independent of the specific values of \(p\) and \(q\) and provides a general formula for the vectors in terms of the given parameters.