91Ó°ÊÓ

Prove the given property. $$(\mathbf{u}-\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=|\mathbf{u}|^{2}-|\mathbf{v}|^{2}$$

Solve the given equation. $$2 \sin ^{2} \theta-\sin \theta-1=0$$

Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and \(\mathbf{j}\). $$|\mathbf{v}|=1, \quad \theta=225^{\circ}$$

Two direction angles of a vector are given. Find the third direction angle, given that it is either obtuse or acute as indicated. (In Exercises 43 and \(44,\) round your answers to the nearest degree.) $$\alpha=60^{\circ}, \quad \beta=50^{\circ} ; \quad \gamma \text { is obtuse }$$

Solve the given equation. $$3 \sin ^{2} \theta-7 \sin \theta+2=0$$

Two direction angles of a vector are given. Find the third direction angle, given that it is either obtuse or acute as indicated. (In Exercises 43 and \(44,\) round your answers to the nearest degree.) $$\alpha=75^{\circ}, \quad \gamma=15^{\circ}$$

Solve the given equation. $$\tan ^{4} \theta-13 \tan ^{2} \theta+36=0$$

Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and \(\mathbf{j}\). $$|\mathbf{v}|=800, \quad \theta=125^{\circ}$$

Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and \(\mathbf{j}\). $$|\mathbf{v}|=4, \quad \theta=10^{\circ}$$

Work The force \(\mathbf{F}=4 \mathbf{i}-7 \mathbf{j}\) moves an object \(4 \mathrm{ft}\) along the \(x\) -axis in the positive direction. Find the work done if the unit of force is the pound.