91Ó°ÊÓ

Solve the given equation. $$\left(\tan ^{2} \theta-4\right)(2 \cos \theta+1)=0$$

Find \(|\mathbf{u}|,|\mathbf{v}|,|2 \mathbf{u}|,\left|\frac{1}{2} \mathbf{v}\right|\) \(|\mathbf{u}+\mathbf{v}|,|\mathbf{u}-\mathbf{v}|,\) and \(|\mathbf{u}|-|\mathbf{v}|.\) $$\mathbf{u}=\langle- 6,6\rangle, \quad \mathbf{v}=\langle- 2,-1\rangle$$

Solve the given equation. $$(\tan \theta-2)\left(16 \sin ^{2} \theta-1\right)=0$$

Find the direction angles of the given vector, rounded to the nearest degree. $$\langle 2,-1,2\rangle$$

Solve the given equation. $$4 \cos ^{2} \theta-4 \cos \theta+1=0$$

Prove the given property. $$(\mathbf{u}+\mathbf{v}) \cdot \mathbf{w}=\mathbf{u} \cdot \mathbf{w}+\mathbf{v} \cdot \mathbf{w}$$

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 \(\mathbf{j}\). $$|\mathbf{v}|=40, \quad \theta=30^{\circ}$$

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.) $$\alpha=\frac{\pi}{3}, \quad \gamma=\frac{2 \pi}{3} ; \beta \text { is acute }$$

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 \(\mathbf{j}\). $$|\mathbf{v}|=50, \quad \theta=120^{\circ}$$

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.) $$\beta=\frac{2 \pi}{3}, \quad \gamma=\frac{\pi}{4} ; \quad \alpha \text { is acute }$$