91Ó°ÊÓ

Velocity A gun with a muzzle velocity of 1200 feet per second is fired at an angle of \(6^{\circ}\) with the horizontal. Find the vertical and horizontal components of the velocity.

What can be said about the vectors \(\mathbf{u}\) and \(\mathbf{v}\) under each condition? (a) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{u}\). (b) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{0}\).

Use vectors to prove that the diagonals of a rhombus are perpendicular.

Prove the following. $$ \|\mathbf{u}-\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}-2 \mathbf{u} \cdot \mathbf{v} $$

In Exercises 81-84, find all solutions of the equation in the interval \([0,2 \pi)\). $$ \sin 2 x-\sqrt{3} \sin x=0 $$

In Exercises 81-84, find all solutions of the equation in the interval \([0,2 \pi)\). $$ \sin 2 x+\sqrt{2} \cos x=0 $$

In Exercises 81-84, find all solutions of the equation in the interval \([0,2 \pi)\). $$ 2 \tan x=\tan 2 x $$

In Exercises 81-84, find all solutions of the equation in the interval \([0,2 \pi)\). $$ \cos 2 x-3 \sin x=2 $$

In Exercises 85-88, find the exact value of the trigonometric function given that \(\sin u=-\frac{12}{13}\) and \(\cos v=\frac{24}{25}\). (Both \(u\) and \(v\) are in Quadrant IV.) $$ \sin (u-v) $$

In Exercises 85-88, find the exact value of the trigonometric function given that \(\sin u=-\frac{12}{13}\) and \(\cos v=\frac{24}{25}\). (Both \(u\) and \(v\) are in Quadrant IV.) $$ \sin (u+v) $$