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4.2E: Projections and Planes Exercises

  2. Last updated
    Jul 25, 2023
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( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercises for 1

solutions

2

Compute u∙v where:

  1. u=[2−13], v=[−111]
  2. u=[12−1], v=u
  3. u=[11−3], v=[2−11]
  4. u=[3−15], v=[6−7−5]
  5. u=[xyz], v=[abc]
  6. u=[abc], v=0
  1. 6
  2. 0
  3. 0

Find the angle between the following pairs of vectors.

  1. u=[103], v=[201]
  2. u=[3−10], v=[−620]
  3. u=[7−13], v=[14−1]
  4. u=[21−1], v=[363]
  5. u=[1−10], v=[011]
  6. u=[034], v=[5√2−7−1]
  1. π or 180∘
  2. π3 or 60∘
  3. 2π3 or 120∘

Find all real numbers x such that:

  1. [2−13] and [x−21] are orthogonal.
  2. [2−11] and [1x2] are at an angle of π3.
  1. 1 or −17

Find all vectors v=[xyz] orthogonal to both:

  1. u1=[−1−32], u2=[011]
  2. u1=[3−12], u2=[201]
  3. u1=[20−1], u2=[−402]
  4. u1=[2−13], u2=[000]
  1. t[−112]
  2. s[120]+t[031]

Find two orthogonal vectors that are both orthogonal to v=[120].

Consider the triangle with vertices P(2,0,−3), Q(5,−2,1), and R(7,5,3).

  1. Show that it is a right-angled triangle.
  2. Find the lengths of the three sides and verify the Pythagorean theorem.
  1. 29+57=86

Show that the triangle with vertices A(4,−7,9), B(6,4,4), and C(7,10,−6) is not a right-angled triangle.

Find the three internal angles of the triangle with vertices:

  1. A(3,1,−2), B(3,0,−1), and C(5,2,−1)
  2. A(3,1,−2), B(5,2,−1), and C(4,3,−3)
  1. A=B=C=π3 or 60∘

Show that the line through P0(3,1,4) and P1(2,1,3) is perpendicular to the line through P2(1,−1,2) and P3(0,5,3).

In each case, compute the projection of u on v.

  1. u=[571], v=[2−13]
  2. u=[3−21], v=[411]
  3. u=[1−12], v=[3−11]
  4. u=[3−2−1], v=[−642]
  1. 1118v
  2. −12v

In each case, write u=u1+u2, where u1 is parallel to v and u2 is orthogonal to v.

  1. u=[2−11], v=[1−13]
  2. u=[310], v=[−214]
  3. u=[2−10], v=[31−1]
  4. u=[3−21], v=[−64−1]
  1. 521[2−1−4]+121[532620]
  2. 2753[6−41]+153[−3226]

Calculate the distance from the point P to the line in each case and find the point Q on the line closest to P.

  1. P(3,2−1)
    line: [xyz]=[213]+t[3−1−2]

  2. P(1,−1,3)
    line: [xyz]=[10−1]+t[314]

  1. 126√5642, Q(7126,1526,3426)

Compute u×v where:

  1. u=[123], v=[112]
  2. u=[3−10], v=[−620]
  3. u=[3−21], v=[11−1]
  4. u=[20−1], v=[147]
  1. [000]
  2. [4−158]

Find an equation of each of the following planes.

  1. Passing through A(2,1,3), B(3,−1,5), and C(1,2,−3).
  2. Passing through A(1,−1,6), B(0,0,1), and C(4,7,−11).
  3. Passing through P(2,−3,5) and parallel to the plane with equation 3x−2y−z=0.
  4. Passing through P(3,0,−1) and parallel to the plane with equation 2x−y+z=3.
  5. Containing P(3,0,−1) and the line [xyz]=[002]+t[101].
  6. [xyz]=[3−12]+t[10−1].
  7. [xyz]=[1−12]+t[111] and [xyz]=[002]+t[1−10].
  8. Containing the lines [xyz]=[310]+t[1−13] and [xyz]=[0−25]+t[21−1].
  9. Each point of which is equidistant from P(2,−1,3) and Q(1,1,−1).
  10. Each point of which is equidistant from P(0,1,−1) and Q(2,−1,−3).
  1. −23x+32y+11z=11
  2. 2x−y+z=5
  3. 2x+3y+2z=7
  4. 2x−7y−3z=−1
  5. x−y−z=3

In each case, find a vector equation of the line.

  1. Passing through P(3,−1,4) and perpendicular to the plane 3x−2y−z=0.
  2. Passing through P(2,−1,3) and perpendicular to the plane 2x+y=1.
  3. Passing through P(0,0,0) and perpendicular to the lines [xyz]=[110]+t[20−1] and [xyz]=[21−3]+t[1−15].
  4. [xyz]=[201]+t[11−2] and
    [xyz]=[55−2]+t[12−3].
  5. Passing through P(2,1,−1), intersecting the line [xyz]=[12−1]+t[301], and perpendicular to that line.
  6. Passing through P(1,1,2), intersecting the line [xyz]=[210]+t[111], and perpendicular to that line.
  1. [xyz]=[2−13]+t[210]
  2. [xyz]=[11−1]+t[111]
  3. [xyz]=[112]+t[41−5]

In each case, find the shortest distance from the point P to the plane and find the point Q on the plane closest to P.

  1. P(2,3,0); plane with equation 5x+y+z=1.
  2. P(3,1,−1); plane with equation 2x+y−z=6.
  1. √63, Q(73,23,−23)
  1. Does the line through P(1,2,−3) with direction vector d=[12−3] lie in the plane 2x−y−z=3? Explain.
  2. Does the plane through P(4,0,5), Q(2,2,1), and R(1,−1,2) pass through the origin? Explain.
  1. Yes. The equation is 5x−3y−4z=0.

Show that every plane containing P(1,2,−1) and Q(2,0,1) must also contain R(−1,6,−5).

Find the equations of the line of intersection of the following planes.

  1. 2x−3y+2z=5 and x+2y−z=4.
  2. 3x+y−2z=1 and x+y+z=5.
  1. (−2,7,0)+t(3,−5,2)

In each case, find all points of intersection of the given plane and the line [xyz]=[1−23]+t[25−1].

x−3y+2z=4 2x−y−z=5 3x−y+z=8 −x−4y−3z=6

  1. None
  2. P(1319,−7819,6519)

Find the equation of all planes:

  1. Perpendicular to the line
    [xyz]=[2−13]+t[213].

  2. Perpendicular to the line
    [xyz]=[10−1]+t[302].

  3. Containing the origin.
  4. Containing P(3,2,−4).
  5. Containing P(1,1,−1) and Q(0,1,1).
  6. Containing P(2,−1,1) and Q(1,0,0).

  7. Containing the line
    [xyz]=[210]+t[1−10].

  8. Containing the line
    [xyz]=[302]+t[1−2−1].

  1. 3x+2z=d, d arbitrary
  2. a(x−3)+b(y−2)+c(z+4)=0; a, b, and c not all zero
  3. ax+by+(b−a)z=a; a and b not both zero
  4. ax+by+(a−2b)z=5a−4b; a and b not both zero

If a plane contains two distinct points P1 and P2, show that it contains every point on the line through P1 and P2.

Find the shortest distance between the following pairs of parallel lines.

  1. [xyz]=[2−13]+t[1−14];
    [xyz]=[101]+t[1−14]

  2. [xyz]=[302]+t[310];
    [xyz]=[−122]+t[310]

  1. √10

Find the shortest distance between the following pairs of nonparallel lines and find the points on the lines that are closest together.

  1. [xyz]=[301]+s[21−3];
    [xyz]=[11−1]+t[101]

  2. [xyz]=[1−10]+s[111];
    [xyz]=[2−13]+t[310]

  3. [xyz]=[31−1]+s[11−1];
    [xyz]=[120]+t[102]

  4. [xyz]=[123]+s[20−1];
    [xyz]=[3−10]+t[110]

  1. √142, A(3,1,2), B(72,−12,3)
  2. √66, A(193,2,13), B(376,136,0)

Show that two lines in the plane with slopes m1 and m2 are perpendicular if and only if
m1m2=−1. [Hint: Example [exa:011343].]

  1. Show that, of the four diagonals of a cube, no pair is perpendicular.
  2. Show that each diagonal is perpendicular to the face diagonals it does not meet.
  1. Consider the diagonal d=[aaa] The six face diagonals in question are ±[a0−a], ±[0a−a], ±[a−a0]. All of these are orthogonal to d. The result works for the other diagonals by symmetry.

Given a rectangular solid with sides of lengths 1, 1, and √2, find the angle between a diagonal and one of the longest sides.

Consider a rectangular solid with sides of lengths a, b, and c. Show that it has two orthogonal diagonals if and only if the sum of two of a2, b2, and c2 equals the third.

The four diagonals are (a,b,c), (−a,b,c), (a,−b,c) and (a,b,−c) or their negatives. The dot products are ±(−a2+b2+c2), ±(a2−b2+c2), and ±(a2+b2−c2).

Let A, B, and C(2,−1,1) be the vertices of a triangle where \longvectAB is parallel to [1−11], \longvectAC is parallel to [20−1], and angle C=90∘. Find the equation of the line through B and C.

If the diagonals of a parallelogram have equal length, show that the parallelogram is a rectangle.

Given v=[xyz] in component form, show that the projections of v on i, j, and k are xi, yj, and zk, respectively.

  1. Can u∙v=−7 if ‖u‖=3 and ‖v‖=2? Defend your answer.
  2. Find u∙v if u=[2−12], ‖v‖=6, and the angle between u and v is 2π3.

Show (u+v)∙(u−v)=‖u‖2−‖v‖2 for any vectors u and v.

  1. Show ‖u+v‖2+‖u−v‖2=2(‖u‖2+‖v‖2) for any vectors u and v.
  2. What does this say about parallelograms?
  1. The sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the four sides.

Show that if the diagonals of a parallelogram are perpendicular, it is necessarily a rhombus. [Hint: Example [exa:011899].]

Let A and B be the end points of a diameter of a circle (see the diagram). If C is any point on the circle, show that AC and BC are perpendicular. [Hint: Express \longvectAB∙(\longvectAB×\longvectAC)=0 and \longvectBC in terms of u=\longvectOA and v=\longvectOC, where O is the centre.]

Show that u and v are orthogonal, if and only if ‖u+v‖2=‖u‖2+‖v‖2.

Let u, v, and w be pairwise orthogonal vectors.

  1. Show that ‖u+v+w‖2=‖u‖2+‖v‖2+‖w‖2.
  2. If u, v, and w are all the same length, show that they all make the same angle with u+v+w.
  1. The angle θ between u and (u+v+w) is given by cosθ=u∙(u+v+w)‖u‖‖u+v+w‖=‖u‖√‖u‖2+‖v‖2+‖w‖2=1√3 because ‖u‖=‖v‖=‖w‖. Similar remarks apply to the other angles.
  1. Show that n=[ab] is orthogonal to every vector along the line ax+by+c=0.
  2. [Hint: If P1 is on the line, project u=\longvectP1P0 on n.]
  1. Let p0, p1 be the vectors of P0, P1, so u=p0−p1. Then u⋅n=p0⋅n – p1⋅n=(ax0+by0)−(ax1+by1)=ax0+by0+c. Hence the distance is

    ‖(u∙n‖n‖2)n‖=|u∙n|‖n‖

Assume u and v are nonzero vectors that are not parallel. Show that w=‖u‖v+‖v‖u is a nonzero vector that bisects the angle between u and v.

Let α, β, and γ be the angles a vector v≠0 makes with the positive x, y, and z axes, respectively. Then cosα, cosβ, and cosγ are called the direction cosines of the vector v.

  1. If v=[abc], show that cosα=a‖v‖, cosβ=b‖v‖, and cosγ=c‖v‖.
  2. Show that cos2α+cos2β+cos2γ=1.
  1. This follows from (a) because ‖v‖2=a2+b2+c2.

Let v≠0 be any nonzero vector and suppose that a vector u can be written as u=p+q, where p is parallel to v and q is orthogonal to v. Show that p must equal the projection of u on v. [Hint: Argue as in the proof of Theorem [thm:011958].]

Let v≠0 be a nonzero vector and let a≠0 be a scalar. If u is any vector, show that the projection of u on v equals the projection of u on av.

  1. Show that the Cauchy-Schwarz inequality |u∙v|≤‖u‖‖v‖ holds for all vectors u and v. [Hint: |cosθ|≤1 for all angles θ.]
  2. [Hint: When is cosθ=±1?]
  3. holds for all numbers x1, x2, y1, y2, z1, and z2.
  4. Show that |xy+yz+zx|≤x2+y2+z2 for all x, y, and z.
  5. Show that (x+y+z)2≤3(x2+y2+z2) holds for all x, y, and z.
  1. Take [x1y1z1]=[xyz] and [x2y2z2]=[yzx] in (c).

Prove that the triangle inequality ‖u+v‖≤‖u‖+‖v‖ holds for all vectors u and v. [Hint: Consider the triangle with u and v as two sides.]

4.2E: Projections and Planes Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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