91Ó°ÊÓ

We can find the solutions of \(\sin x=0.3\) algebraically. (a) First we find the solutions in the interval \([0,2 \pi) .\) We get one such solution by taking \(\sin ^{-1}\) to get \(x \approx\) _______. The other solution in this interval is \(x \approx\) _______. (b) We find all solutions by adding multiples of _______ to the solutions in \([0,2 \pi) .\) The solutions are \(x \approx\) _______ and \(x \approx\) _______.

Two vectors \(u\) and \(v\) are given. (a) Find a vector orthogonal (perpendicular) to both \(\mathbf{u}\) and \(\mathbf{v}\). (b) Find a unit vector orthogonal (perpendicular) to both \(\mathbf{u}\) $$\mathbf{u}=3 \mathbf{j}+5 \mathbf{k}, \quad \mathbf{v}=-\mathbf{i}+2 \mathbf{k}$$

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 \(\mathbf{j}\). $$|\mathbf{v}|=1, \quad \theta=225^{\circ}$$

A unit vector is a vector of magnitude 1. Mul. tiplying a vector by a scalar changes its magnitude but not its direction. (a) If a vector \(\mathbf{v}\) has magnitude \(m,\) what scalar multiple of \(\mathbf{v}\) has magnitude 1 (that is, is a unit vector)? (b) Multiply each of the following vectors by an appropriate scalar to change them into unit vectors: $$\langle 1,-2,2\rangle\langle- 6,8,-10\rangle \quad\langle 6,5,9\rangle$$

A tetrahedron is a solid with four triangular faces, four vertices, and six edges, as shown in the figure. In a regular tetrahedron the edges are all of the same length. Consider the tetrahedron with vertices \(A(1,0,0), B(0,1,0), C(0,0,1),\) and \(D(1,1,1)\) (a) Show that the tetrahedron is regular. (b) The center of the tetrahedron is the point \(E\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)\) (the "average" of the vertices). Find the angle between the vectors that join the center to any two of the vertices (for instance, \(\angle A E B\) ). This angle is called the central angle of the tetrahedron.

Components of a Force A man pushes a lawn mower with a force of 30 lb exerted at an angle of \(30^{\circ}\) to the ground. Find the horizontal and vertical components of the force.