Exercises for Section 11.2

Question 11.48

Calculate \((3{\bf i}+2{\bf j}+{\bf k})\,{ \cdot}\,({\bf i}+2{\bf j}-{\bf k})\).

Question 11.49

Calculate \({\bf a}\,{ \cdot}\,{\bf b}\), where \({\bf a}=2{\bf i}+10{\bf j}-12{\bf k}\) and \({\bf b}=-3{\bf i}+4{\bf k}\).

Question 11.50

Find the angle between \(7{\bf j}+19{\bf k}\) and \(-2{\bf i}-{\bf j}\) (to the nearest degree).

Question 11.51

Compute \({\bf u}\,{ \cdot}\,{\bf v}\), where \({\bf u}=\sqrt{3}{\bf i}-315{\bf j}+22{\bf k}\) and \({\bf v}={\bf u}/\|{\bf u}\|\).

Question 11.52

Is \(\|8{\bf i}-12{\bf k}\| \,{ \cdot}\, \|6{\bf j}+{\bf k}\|-|(8{\bf i}-12{\bf k}) \,{ \cdot}\, (6{\bf j}+{\bf k})|\) equal to zero? Explain.

In Exercises 6 to 11, compute \(\|{\bf u}\|,\|{\bf v}\|,\) and \({\bf u}\,{ \cdot}\,{\bf v}\) for the given vectors in \({\mathbb R}^3\).

Question 11.53

\({\bf u}=15{\bf i}-2{\bf j}+4{\bf k},{\bf v}=\pi {\bf i}+3{\bf j}-{\bf k}\)

Question 11.54

\({\bf u}=2{\bf j}-{\bf i},{\bf v}=-{\bf j}+{\bf i}\)

Question 11.55

\({\bf u}=5{\bf i}-{\bf j}+2{\bf k},{\bf v}={\bf i}+{\bf j}-{\bf k}\)

Question 11.56

\({\bf u}=-{\bf i}+3{\bf j}+{\bf k},{\bf v}=-2{\bf i}-3{\bf j}-7{\bf k}\)

Question 11.57

\({\bf u}=-{\bf i}+3{\bf k},{\bf v}=4{\bf j}\)

Question 11.58

\({\bf u}=-{\bf i}+2{\bf j}-3{\bf k},{\bf v}=-{\bf i}-3{\bf j}+4{\bf k}\)

Question 11.59

Let \(\textbf{v}=(2,3)\). Suppose \(\textbf{w} \in \mathbb{R}^2\) is perpendicular to \(\textbf{v}\), and that \(\| \textbf{w} \| =5\). This determines \(\textbf{w}\) up to sign. Find one such \(\textbf{w}\).

Question 11.60

Find \(b\) and \(c\) so that \((5, b, c)\) is orthogonal to both (1, 2, 3) and (1, \(-\)2, 1).

Question 11.61

Let \(\textbf{v}_1=(0, 3, 0), \textbf{v}_2=(2, 2, 0), \textbf{v}_3=(1, 1, 3)\). These three vectors with their tails at the origin determine a parallelepiped \(P\).

  • (a) Draw \(P\).
  • (b) Determine the length of the main diagonal (from the origin to its opposite vertex).

Question 11.62

What is the geometric relation between the vectors \(\textbf{v}\) and \(\textbf{w}\) if \(\textbf{v} \,{\cdot}\, \textbf{w} = -\|\textbf{v}\| \ \|\textbf{w}\|\)?

Question 11.63

Normalize the vectors in Exercises 6 to 8. (Only the solution corresponding to Exercise 7 is in the Student Guide.)

Question 11.64

Find the angle between the vectors in Exercises 9 to 11. If necessary, express your answer in terms of \(\cos^{-1}\).

Question 11.65

Find all values of \(x\) such that \((x, 1, x)\) and \((x, -6, 1)\) are orthogonal.

Question 11.66

Find all values of \(x\) such that \((7, x, -10)\) and \((3, x, x)\) are orthogonal.

Question 11.67

Find the projection of \({\bf u}=-{\bf i}+{\bf j}+{\bf k}\) onto \({\bf v}=2{\bf i}+{\bf j}-3{\bf k}\).

Question 11.68

Find the projection of \({\bf v}=2{\bf i}+{\bf j}-3{\bf k}\) onto \({\bf u}=-{\bf i}+{\bf j}+{\bf k}.\)

Question 11.69

What restrictions must be made on the scalar \(b\) so that the vector \(2{\bf i}+ b{\bf j}\) is orthogonal to (a) \(-3{\bf i}+2{\bf j}+{\bf k}\) and (b) \({\bf k}\)?

Question 11.70

Vectors \(\textbf{v}\) and \(\textbf{w}\) are sides of an equilateral triangle whose sides have length 1. Compute \(\textbf{v} \,{\cdot}\, \textbf{w}\).

Question 11.71

Let \(\textbf{b}=(3,1,1)\) and \(P\) be the plane through the origin given by \(x+y+2z=0\).

  • (a) Find an orthogonal basis for \(P\). That is, find two nonzero orthogonal vectors \(\textbf{v}_1, \textbf{v}_2 \in P\).
  • (b) Find the orthogonal projection of \(\textbf{b}\) onto \(P\). That is, find \(\hbox{Proj}_{\textbf{v}_1} \textbf{b} + \text{Proj}_{\textbf{v}_2} \textbf{b}\).

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Question 11.72

Find two nonparallel vectors both orthogonal to (1, 1, 1).

Question 11.73

Find the line through \((3,1,-2)\) that intersects and is perpendicular to the line \(x=-1+t,y=-2+t,z=-1+t\). [HINT: If \((x_0,y_0,z_0)\) is the point of intersection, find its coordinates.]

Question 11.74

Using the dot product, prove the converse of the Pythagorean theorem. That is, show that if the lengths of the sides of a triangle satisfy \(a^2+b^2=c^2\), then the triangle is a right triangle.

Question 11.75

For \(\textbf{v}=(v_1, v_2, v_3)\) let \(\alpha, \beta, \gamma\) denote the angles between \(\textbf{v}\) and the \(x, y,\) and \(z\) axes, respectively. Show that \(\cos^2 \alpha+\cos^2 \beta+\cos^2 \gamma = 1\).

Question 11.76

A ship at position (1, 0) on a nautical chart (with north in the positive \(y\) direction) sights a rock at position (2, 4). What is the vector joining the ship to the rock? What angle \(\theta\) does this vector make with due north? (This is called the bearing of the rock from the ship.)

Question 11.77

Suppose that the ship in Exercise 29 is pointing due north and traveling at a speed of \(4\) knots relative to the water. There is a current flowing due east at 1 knot. The units on the chart are nautical miles; \(1 \hbox{ knot} =1 \hbox{ nautical}\) mile per hour.

  • (a) If there were no current, what vector \({\bf u}\) would represent the velocity of the ship relative to the sea bottom?
  • (b) If the ship were just drifting with the current, what vector {\bf v} would represent its velocity relative to the sea bottom?
  • (c) What vector \({\bf w}\) represents the total velocity of the ship?
  • (d) Where would the ship be after 1 hour?
  • (e) Should the captain change course?
  • (f) What if the rock were an iceberg?

Question 11.78

An airplane is located at position \((3, 4, 5)\) at noon and traveling with velocity \(400{\bf i}+500{\bf j}-{\bf k}\) kilometers per hour. The pilot spots an airport at position (23, 29, 0).

  • (a) At what time will the plane pass directly over the airport? (Assume that the plane is flying over flat ground and that the vector \({\bf k}\) points straight up.)
  • (b) How high above the airport will the plane be when it passes?

Question 11.79

The wind velocity \({\bf v}_1\) is 40 miles per hour (mi/h) from east to west while an airplane travels with air speed \({\bf v}_2\) of 100 mi/h due north. The speed of the airplane relative to the ground is the vector sum \({\bf v}_1+{\bf v}_2\).

  • (a) Find \({\bf v}_1+{\bf v}_2.\)
  • (b) Draw a figure to scale.

Question 11.80

A force of 50 lb is directed 50\(^\circ\) above horizontal, pointing to the right. Determine its horizontal and vertical components. Display all results in a figure.

Question 11.81

Two persons pull horizontally on ropes attached to a post, the angle between the ropes being 60\(^\circ\). Person A pulls with a force of 150 lb, while person B pulls with a force of 110 lb.

  • (a) The resultant force is the vector sum of the two forces. Draw a figure to scale that graphically represents the three forces.
  • (b) Using trigonometry, determine formulas for the vector components of the two forces in a conveniently chosen coordinate system. Perform the algebraic addition, and find the angle the resultant force makes with A.

Question 11.82

A 1-kilogram (1-kg) mass located at the origin is suspended by ropes attached to the two points (1, 1, 1) and \((-1,-1,1)\). If the force of gravity is pointing in the direction of the vector \(-{\bf k}\), what is the vector describing the force along each rope? [HINT: Use the symmetry of the problem. A 1-kg mass weighs 9.8 newtons (N).]

Question 11.83

Suppose that an object moving in direction \({\bf i}+{\bf j}\) is acted on by a force given by the vector \(2{\bf i}+{\bf j}\). Express this force as the sum of a force in the direction of motion and a force perpendicular to the direction of motion.

Question 11.84

A force of 6 N makes an angle of \(\pi/4\) radian with the \(y\) axis, pointing to the right. The force acts against the movement of an object along the straight line connecting (1, 2) to (5, 4).

  • (a) Find a formula for the force vector \({\bf F}\).
  • (b) Find the angle \(\theta\) between the displacement direction \({\bf D}=(5-1) {\bf i}+(4-2){\bf j}\) and the force direction F.
  • (c) The work done is \({\bf F}\,{ \cdot}\,{\bf D}\), or, equivalently, \(\|{\bf F}\|\|{\bf D}\|\cos \theta\). Compute the work from both formulas and compare.

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Question 11.85

Show that in any parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals.

Question 11.86

Using vectors, show that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.