Crosswinds A small plane is flying horizontally due east

Collinear points Determine the values of x and y such that the points (1, 2, 3), (4, 7, 1), and (x, y, 2) are collinear (lie on a line).

Medians of a triangle—coordinate free Assume that u, v, and w are vectors in \(\mathbf{R}^3\) that form the sides of a triangle (see figure). Use the following steps to prove that the medians intersect at a point that divides the median in a 2:1 ratio. The proof does not use a coordinate system. a. Show that u + v + w = 0. b. Let \(\mathbf{M}_{1}\) be the median vector from the midpoint of u to the opposite vertex. Define \(\mathbf{M}_{2}\) and \(\mathbf{M}_{3}\) similarly. Using the geometry of vector addition show that \(\mathbf{M}_{1}=\mathbf{u} / 2+\mathbf{v}\). Find analogous expressions for \(\mathbf{M}_{2}\) and \(\mathbf{M}_{3}\). c. Let a, b, and c be the vectors from O to the points one-third of the way along \(\mathbf{M}_{1}\), \(\mathbf{M}_{2}\), and \(\mathbf{M}_{3}\), respectively. Show that a = b = c = (u - w)/3. d. Conclude that the medians intersect at a point that divides each median in a 2:1 ratio.

Planes Sketch the plane parallel to the xy-plane through (2, 4, 2) and find its equation.

Unit vectors and Magnitude Consider the following points P and Q. a. Find \(\overrightarrow{P Q}\) and state your answer in two forms: \(\langle a, b, c\rangle\) and ai + bj + ck. b. Find the magnitude of \(\overrightarrow{P Q}\). c. Find two unit vectors parallel to \(\overrightarrow{P Q}\). P(5, 11, 12), Q(1, 14, 13)

Unit vectors and Magnitude Consider the following points P and Q. a. Find \(\overrightarrow{P Q}\) and state your answer in two forms: \(\langle a, b, c\rangle\) and ai + bj + ck. b. Find the magnitude of \(\overrightarrow{P Q}\). c. Find two unit vectors parallel to \(\overrightarrow{P Q}\). P(a, b, c), Q(1, 1, ?1) (a, b, and c are real numbers)

Identifying sets Give a geometric description of the following sets of points. \(x^{2}+y^{2}+z^{2}-8 x-14 y-18 z \leq 65\)

Identifying sets Give a geometric description of the following sets of points. \(x^{2}+y^{2}+z^{2}-2 y-4 z-4=0\)

Let \(\mathbf{u}=\langle 3,5,-7\rangle\) and \(\mathbf{v}=\langle 6,-5,1\rangle\). Evaluate u + v and 3u - v.

Medians of a triangle-with coordinates In contrast to the proof in Exercise 67, we now use coordinates and position vectors to prove the same result. Without loss of generality, let \(P\left(x_{1}, y_{1}, 0\right)\) and \(Q\left(x_{2}, y_{2}, 0\right)\) be two points in the xy-plane and let \(R\left(x_{3}, y_{3}, z_{3}\right)\) be a third point, such that P, Q, and R do not lie on a line. Consider \(\triangle P Q R\). a. Let \(M_{1}\) be the midpoint of the side PQ. Find the coordinates of \(M_{1}\) and the components of the vector \(\overrightarrow{R M}_{1}\). b. Find the vector \(\overrightarrow{O Z_{1}}\) from the origin to the point \(Z_{1}\) two-thirds of the way along \(\overrightarrow{R M_{1}}\). c. Repeat the calculation of part (b) with the midpoint \(M_{2}\) of RQ and the vector \(\overrightarrow{P M}_{2}\) to obtain the vector \(\overrightarrow{O Z}_{2}\). d. Repeat the calculation of part (b) with the midpoint \(M_{3}\) of PR and the vector \(\overrightarrow{Q M}_{3}\) to obtain the vector \(\overrightarrow{O Z}_{3}\). e. Conclude that the medians of \(\triangle P Q R\) intersect at a point. Give the coordinates of the point. f. With P(2, 4, 0), Q(4, 1, 0), R(6, 3, 4), find the point at which the medians of \(\triangle P Q R\) intersect.

What position vector is equal to the vector from (3, 5, ?2) to (0, ?6, 3)?

Plotting points in \(\mathbf{R}^3\) For each point P(x, y, z) given below, let A(x, y, 0), B(x, 0, z), and C(0, y, z) be points in the xy-, xz- and yz-planes respectively. Plot and label the points A, B, C, and P in \(\mathbf{R}^3\). a. P(2, 2, 4) b. P(1, 2, 5) c. P(?2, 0, 5)

Vector operations For the given vectors u and v, evaluate the following expressions. a. 3u + 2v b. 4u ? v c. |u + 3v| \(\mathbf{u}=\langle 5,1,3 \sqrt{2}\rangle, \mathbf{v}=\langle 2,0,7 \sqrt{2}\rangle\)

Midpoints and spheres Find an equation of the sphere passing through P(1, 0, 5) and Q(2, 3, 9) with its center at the midpoint of PQ.

Express the vector from P(-1, -4, 6) to Q(1, 3, -6) as a position vector in terms of i, j, and k.

Unit vectors and Magnitude Consider the following points P and Q. a. Find \(\overrightarrow{P Q}\) and state your answer in two forms: \(\langle a, b, c\rangle\) and ai + bj + ck. b. Find the magnitude of \(\overrightarrow{P Q}\). c. Find two unit vectors parallel to \(\overrightarrow{P Q}\). P(3, 8, 12), Q(3, 9, 11)

Equation of a sphere For constants a, b, c, and d, show that the equation \(x^{2}+y^{2}+z^{2}-2 a x-2 b y-2 c z=d\) describes a sphere centered at (a, b, c) with radius r, where \(r^{2}=d+a^{2}+b^{2}+c^{2}\), provided \(d+a^{2}+b^{2}+c^{2}>0\).

Identifying sets Give a geometric description of the following sets of points. \(x^{2}+y^{2}-14 y+z^{2} \geq-13\)

Spheres and balls Find an equation or inequality that describes the following objects. A ball with center (?2, 0, 4) and radius 1

Identifying sets Give a geometric description of the following sets of points. \(x^{2}+y^{2}+z^{2}-8 x+14 y-18 z \geq 65\)

Points in \(\mathbf{R}^{3}\) Find the coordinates of the vertices A, B, and C of the following rectangular boxes.

Vector operations For the given vectors u and v, evaluate the following expressions. a. 3u + 2v b. 4u ? v c. |u + 3v| \(\mathbf{u}=\langle 1,3,0\rangle, \mathbf{v}=\langle 3,0,2\rangle\)

Sketching planes Sketch the following planes in the window [0, 5] x [0, 5] x [0, 5]. The plane that passes through (2, 0, 0), (0, 3, 0), and (0, 0, 4)

Describe the plane x = 4.

What is the magnitude of a vector joining two points \(P\left(x_{1}, y_{1}, z_{1}\right)\) and \(Q\left(x_{2}, y_{2}, z_{2}\right)\)?

Sketching planes Sketch the following planes in the window [0, 5] x [0, 5] x [0, 5]. z = y

Vector operations For the given vectors u and v, evaluate the following expressions. a. 3u + 2v b. 4u ? v c. |u + 3v| \(\mathbf{u}=\langle-1,1,0\rangle, \mathbf{v}=\langle 2,-4,1\rangle\)

Vector operations For the given vectors u and v, evaluate the following expressions. a. 3u + 2v b. 4u ? v c. |u + 3v| \(\mathbf{u}=\langle-7,5,1\rangle, \mathbf{v}=\langle-2,4,0\rangle\)

Unit vectors and Magnitude Consider the following points P and Q. a. Find \(\overrightarrow{P Q}\) and state your answer in two forms: \(\langle a, b, c\rangle\) and ai + bj + ck. b. Find the magnitude of \(\overrightarrow{P Q}\). c. Find two unit vectors parallel to \(\overrightarrow{P Q}\). P(-3, 1, 0), Q(-3, -4, 1)

Planes Sketch the plane parallel to the yz-plane through (2, 4, 2) and find its equation.

Spheres and balls Find an equation or inequality that describes the following objects. A sphere with center (1, 2, 0) passing through the point (3, 4, 5)

Plotting points in \(\mathbf{R}^3\) For each point P(x, y, z) given below, let A(x, y, 0), B(x, 0, z), and C(0, y, z) be points in the xy-, xz- and yz-planes respectively. Plot and label the points A, B, C, and P in \(\mathbf{R}^3\). a. P(-3, 2, 4) b. P(4, -2, -3) c. P(-2, -4, -3)

Parallel vectors of varying lengths Find vectors parallel to v of the given length. \(\mathbf{v}=\overrightarrow{P Q}\) with P(3, 4, 0) and Q(2, 3, 1); length = 3

Combined force An object at the origin is acted on by the forces \(\mathbf{F}_{1}=20 \mathbf{i}-10 \mathbf{j}\), \(\mathbf{F}_{2}=30 \mathbf{j}+10 \mathbf{k}\), and \(\mathbf{F}_{3}=40 \mathbf{j}+20 \mathbf{k}\). Find the magnitude of the combined force and describe the approximate direction of the force.

Midpoints and spheres Find an equation of the sphere passing through P(-4, 2, 3) and Q(0, 2, 7) with its center at the midpoint of PQ.

Maintaining equilibrium An object is acted upon by the forces \(\mathbf{F}_{1}=\langle 10,6,3\rangle\) and \(\mathbf{F}_{2}=\langle 0,4,9\rangle\). Find the force \(\mathbf{F}_{3}\) that must act on the object so that the sum of the forces is zero.

Sketching planes Sketch the following planes in the window [0, 5] x [0, 5] x [0, 5]. z = 3

Which point is farther from the origin, (3, ?1, 2) or (0, 0, ?4)?

Unit vectors and Magnitude Consider the following points P and Q. a. Find \(\overrightarrow{P Q}\) and state your answer in two forms: \(\langle a, b, c\rangle\) and ai + bj + ck. b. Find the magnitude of \(\overrightarrow{P Q}\). c. Find two unit vectors parallel to \(\overrightarrow{P Q}\). P(0, 0, 2), Q(?2, 4, 0)

Identifying sets Give a geometric description of the following sets of points. \(x^{2}+y^{2}+z^{2}-6 x+6 y-8 z-2=0\)

What is the y-coordinate of all points in the xz-plane?

Parallel vectors of varying lengths Find vectors parallel to v of the given length. \(\mathbf{v}=\overrightarrow{P Q}\) with P(1, 0, 1) and Q(2, -1, 1); length = 3

Points in \(\mathbf{R}^{3}\) Find the coordinates of the vertices A, B, and C of the following rectangular boxes. Assume all the edges have the same length.

Four-cable load A 500-lb load hangs from four cables of equal length that are anchored at the points \((\pm 2,0,0)\) and \((0, \pm 2,0)\). The load is located at (0, 0, -4). Find the vectors describing the forces on the cables due to the load.

Lengths of the diagonals of a box A fisherman wants to know if his fly rod will fit in a rectangular 2 ft x 3 ft x 4 ft packing box. What is the longest rod that fits in this box?

Parallel vectors of varying lengths Find vectors parallel to v of the given length. \(\mathbf{v}=\langle 3,-2,6\rangle\); length = 10

Possible parallelograms The points O(0, 0, 0), P(1, 4, 6), and Q(2, 4, 3) lie at three vertices of a parallelogram. Find all possible locations of the fourth vertex.

Identifying sets Give a geometric description of the following sets of points. \(x^{2}+y^{2}-14 y+z^{2} \leq-13\)

Three-cable load A 500-lb load hangs from three cables of equal length that are anchored at the points (-2, 0, 0), \((1, \sqrt{3}, 0)\), and \((1,-\sqrt{3}, 0)\). The load is located at \((0,0,-2 \sqrt{3})\). Find the vectors describing the forces on the cables due to the load.

Spheres and balls Find an equation or inequality that describes the following objects. A ball with center (0, ?2, 6) with the point (1, 4, 8) on its boundary

Sketching planes Sketch the following planes in the window [0, 5] x [0, 5] x [0, 5]. y = 2

Spheres and balls Find an equation or inequality that describes the following objects. A sphere with center (1, 2, 3) and radius 4

Midpoint formula Prove that the midpoint of the line segment joining \(P\left(x_{1}, y_{1}, z_{1}\right)\) and \(Q\left(x_{2}, y_{2}, z_{2}\right)\) is \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)\).

Points in \(\mathbf{R}^{3}\) Find the coordinates of the vertices A, B, and C of the following rectangular boxes.

The amazing quadrilateral property-coordinate free The points P, Q, R, and S, joined by the vectors u, v, w, x are the vertices of a quadrilateral in \(\mathbf{R}^{3}\). The four points needn't lie in a plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system. a. Use vector addition to show that u + v = w + x. b. Let m be the vector that joins the midpoints of PQ and QR. Show that m = (u + v)/2. c. Let n be the vector that joins the midpoints of PS and SR. Show that n = (x + w) / 2. d. Combine parts (a), (b), and (c) to conclude that m = n. e. Explain why part (d) implies that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram.

Sketching planes Sketch the following planes in the window [0, 5] x [0, 5] x [0, 5]. x = 2

Sets of points Describe with a sketch the sets of points (x, y, z) satisfying the following equations. (x + 1)(y - 3) = 0

Sketching planes Sketch the following planes in the window [0, 5] x [0, 5] x [0, 5]. The plane parallel to the xz-plane containing the point (1, 2, 3)

Unit vectors and Magnitude Consider the following points P and Q. a. Find \(\overrightarrow{P Q}\) and state your answer in two forms: \(\langle a, b, c\rangle\) and ai + bj + ck. b. Find the magnitude of \(\overrightarrow{P Q}\). c. Find two unit vectors parallel to \(\overrightarrow{P Q}\). P(1, 5, 0), Q(3, 11, 2)

Explain how to plot the point (3, -2, 1) in \(\mathbf{R}^{3}\).

Sets of points Describe with a sketch the sets of points (x, y, z) satisfying the following equations. \(x^{2} y^{2} z^{2}>0\)

Crosswinds A small plane is flying horizontally due east in calm air at 250 mi/hr when it is hit by a horizontal crosswind blowing southwest at 50 mi/hr and a 30-mi/hr updraft. Find the resulting speed of the plane and describe with a sketch the approximate direction of the velocity relative to the ground.

Parallel vectors of varying lengths Find vectors parallel to v of the given length. \(\mathbf{v}=\langle 6,-8,0\rangle\); length = 20

Collinear points Determine whether the points P, Q, and R are collinear (lie on a line) by comparing \(\overrightarrow{P Q}\) and \(\overrightarrow{P R}\). If the points are collinear, determine which point lies between the other two points. a. P(1, 6, -5), Q(2, 5, -3), R(4, 3, 1) b. P(1, 5, 7), Q(5, 13, -1), R(0, 3, 9) c. P(1, 2, 3), Q(2, -3, 6), R(3, -1, 9) d. P(9, 5, 1), Q(11, 18, 4), R(6, 3, 0)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Suppose u and v both make a \(45^{\circ}\) angle with w in \(\mathbf{R}^{3}\). Then, u + v makes a \(45^{\circ}\) angle with w. b. Suppose u and v both make a \(90^{\circ}\) angle with w in \(\mathbf{R}^{3}\). Then, u + v can never make a \(90^{\circ}\) angle with w. c. i + j + k = 0 d. The intersection of the planes x = 1, y = 1, and z = 1 is a point.

Sets of points Describe with a sketch the sets of points (x, y, z) satisfying the following equations. y - z = 0

Points in \(\mathbf{R}^{3}\) Find the coordinates of the vertices A, B, and C of the following rectangular boxes.

Submarine course A submarine climbs at an angle of \(30^{\circ}\) above the horizontal with a heading to the northeast. If its speed is 20 knots, find the components of the velocity in the east, north, and vertical directions.

Forces on an inclined plane An object on an inclined plane does not slide provided the component of the object’s weight parallel to the plane \(|\mathbf{W}_\mathrm {par}|\) is less than or equal to the magnitude of the opposing frictional force \(|\mathbf{F}_\mathrm {f}|\). The magnitude of the frictional force, in turn, is proportional to the component of the object’s weight perpendicular to the plane \(|\mathbf{W}_\mathrm {perp}|\) (see figure). The constant of proportionality is the coefficient of static-friction, \(\mu\). a. Suppose a 100-lb block rests in a plane that is tilted at an angle of \(\theta=20^{\circ}\) to the horizontal. Find \(\left|\mathbf{W}_{\text {par }}\right|\) and \(\left|\mathbf{W}_{\text {perp }}\right|\). b. The condition for the block not sliding is \(\left|\mathbf{W}_{\text {par }}\right| \leq \mu\left|\mathbf{W}_{\text {perp }}\right|\). If \(\mu=0.65\), does the block slide? c. What is the critical angle above which the block slides?

Diagonals of parallelograms Two sides of a parallelogram are formed by the vectors u and v. Prove that the diagonals of the parallelogram are u + v and u - v.

The amazing quadrilateral property-with coordinates Prove the quadrilateral property in Exercise 69 assuming the coordinates of P, Q, R, and S are \(P\left(x_{1}, y_{1}, 0\right), Q\left(x_{2}, y_{2}, 0\right), R\left(x_{3}, y_{3}, 0\right)\), and \(S\left(x_{4}, y_{4}, z_{4}\right)\), where we assume that P, Q, and R lie in the xy-plane without loss of generality.