91Ó°ÊÓ

In Exercises \(1-4\), plot the points on the same three-dimensional coordinate system. (a) \((2,1,3)\) (b) \((-1,2,1)\)

The figure shows the graph of a line given by the parametric equations. (a) Draw an arrow on the line to indicate its orientation. To print an enlarged copy of the graph, select the MathGraph button. (b) Find the coordinates of two points, \(P\) and \(Q\), on the line. Determine the vector \(\overrightarrow{P Q} .\) What is the relationship between the components of the vector and the coefficients of \(t\) in the parametric equations? Why is this true? (c) Determine the coordinates of any points of intersection with the coordinate planes. If the line does not intersect a coordinate plane, explain why.\(x=2-3 t\) \(y=2\) \(z=1-t\)

Find sets of (a) parametric equations and (b) symmetric equations of the line through the point parallel to the given vector or line. (For each line, write the direction numbers as integers.)\((-2,0,3) \quad \mathbf{v}=2 \mathbf{i}+4 \mathbf{j}-2 \mathbf{k}\)

The initial and terminal points of a vector \(\mathbf{v}\) are given. (a) Sketch the given directed line segment, (b) write the vector in component form, and (c) sketch the vector with its initial point at the origin. $$ (10,2) \quad(6,-1) $$

In Exercises 13-24, determine the location of a point \((x, y, z)\) that satisfies the condition(s). \(z=6\)

Describe and sketch the surface. $$ y^{2}-z^{2}=4 $$

Describe and sketch the surface. $$ z-\sin y=0 $$

The vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. $$ (-3,0,0),(0,0,0),(1,2,3) $$

The vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. $$ (2,-3,4),(0,1,2),(-1,2,0) $$

The vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. $$ (2,-7,3),(-1,5,8),(4,6,-1) $$