91Ó°ÊÓ

Find a set of parametric equations for the line of intersection of the planes.\(6 x-3 y+z=5\) \(-x+y+5 z=5\)

Use vectors to prove that a parallelogram is a rectangle if and only if its diagonals are equal in length.

Think About It Consider two forces of equal magnitude acting on a point. (a) If the magnitude of the resultant is the sum of the magnitudes of the two forces, make a conjecture about the angle between the forces. (b) If the resultant of the forces is 0, make a conjecture about the angle between the forces. (c) Can the magnitude of the resultant be greater than the sum of the magnitudes of the two forces? Explain.

\mathrm{\\{} B o n d ~ A n g l e ~ C o n s i d e r ~ a ~ r e g u l a r ~ t e t r a h e d r o n ~ w i t h ~ v e r t i c e s ~ \((0,0,0),(k, k, 0),(k, 0, k)\), and \((0, k, k)\), where \(k\) is a positive real number. (a) Sketch the graph of the tetrahedron. (b) Find the length of each edge. (c) Find the angle between any two edges. (d) Find the angle between the line segments from the centroid \((k / 2, k / 2, k / 2)\) to two vertices. This is the bond angle for a molecule such as \(\mathrm{CH}_{4}\) or \(\mathrm{PbCl}_{4}\), where the structure of the molecule is a tetrahedron.

Consider the vectors \(\mathbf{u}=\langle\cos \alpha, \sin \alpha, 0\rangle\) and \(\mathbf{v}=\langle\cos \beta, \sin \beta, 0\rangle\) where \(\alpha>\beta\). Find the dot product of the vectors and use the result to prove the identity \(\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta\)

Find the point(s) of intersection (if any) of the plane and the line. Also determine whether the line lies in the plane.\(5 x+3 y=17, \quad \frac{x-4}{2}=\frac{y+1}{-3}=\frac{z+2}{5}\)

Use vectors to find the points of trisection of the line segment with endpoints \((1,2)\) and \((7,5)\).

Navigation A plane flies at a constant groundspeed of 400 miles per hour due east and encounters a 50 -mile-per-hour wind from the northwest. Find the airspeed and compass direction that will allow the plane to maintain its groundspeed and eastward direction.

Sketch the vector \(v\) and write its component form. \(\mathrm{v}\) lies in the \(x z\) -plane, has magnitude 5 , and makes an angle of \(45^{\circ}\) with the positive \(z\) -axis.

Geometry Using vectors, prove that the line segment joining the midpoints of two sides of a triangle is parallel to, and onehalf the length of, the third side.