91Ó°ÊÓ

Find parametric equations for the lines in Exercises \(1-12\) The line through the point \(P(3,-4,-1)\) parallel to the vector \(\mathbf{i}+\mathbf{j}+\mathbf{k}\)

Find parametric equations for the lines in Exercises 1-12. The \(z\) -axis

Rectangle Find the measures of the angles between the diagonals of the rectangle whose vertices are \(A=(1,0), B=(0,3)\) \(C=(3,4),\) and \(D=(4,1)\)

Diagonals of a rhombus Show that the diagonals of a rhombus (parallelogram with sides of equal length) are perpendicular.

Perpendicular diagonals Show that squares are the only rectangles with perpendicular diagonals.

Parallel and perpendicular vectors Let \(\mathbf{u}=5 \mathbf{i}-\mathbf{j}+\mathbf{k}, \mathbf{v}=$$\mathbf{j}-5 \mathbf{k}, \mathbf{w}=-15 \mathbf{i}+3 \mathbf{j}-3 \mathbf{k} .\) Which vectors, if any, are (a) perpendicular? (b) Parallel? Give reasons for your answers.

Find the areas of the parallelograms whose vertices are given in Exercises \(35-40 .\) $$ A(-1,2), \quad B(2,0), \quad C(7,1), \quad D(4,3) $$