4.1E: Vectors and Lines Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
Exercises for 1
solutions
2
Compute โvโ if v equals:
[2โ12] [1โ12] [10โ1] [โ102] 2[1โ12] โ3[112]
- โ6
- โ5
- 3โ6
Find a unit vector in the direction of:
[7โ15] [โ2โ12]
- 13[โ2โ12]
-
Find a unit vector in the direction from
[3โ14] to [135]. - If uโ 0, for which values of a is au a unit vector?
Find the distance between the following pairs of points.
[3โ10] and [2โ11] [2โ12] and [201] [โ352] and [133] [40โ2] and [320]
- โ2
- 3
Use vectors to show that the line joining the midpoints of two sides of a triangle is parallel to the third side and half as long.
Let A, B, and C denote the three vertices of a triangle.
- If E is the midpoint of side BC, show that
\longvectAE=12(\longvectAB+\longvectAC)
- If F is the midpoint of side AC, show that
\longvectFE=12\longvectAB
- \longvectFE=\longvectFC+\longvectCE=12\longvectAC+12\longvectCB=12(\longvectAC+\longvectCB)=12\longvectAB
Determine whether u and v are parallel in each of the following cases.
- u=[โ3โ63]; v=[510โ5]
- u=[3โ63]; v=[โ12โ1]
- u=[101]; v=[โ101]
- u=[20โ1]; v=[โ804]
- Yes
- Yes
Let p and q be the vectors of points P and Q, respectively, and let R be the point whose vector is p+q. Express the following in terms of p and q.
\longvectQP \longvectQR \longvectRP \longvectRO where O is the origin
- p
- โ(p+q).
In each case, find \longvectPQ and โ\longvectPQโ.
- P(1,โ1,3), Q(3,1,0)
- P(2,0,1), Q(1,โ1,6)
- P(1,0,1), Q(1,0,โ3)
- P(1,โ1,2), Q(1,โ1,2)
- P(1,0,โ3), Q(โ1,0,3)
- P(3,โ1,6), Q(1,1,4)
- [โ1โ15], โ27
- [000], 0
- [โ222], โ12
In each case, find a point Q such that \longvectPQ has (i) the same direction as v; (ii) the opposite direction to v.
- P(โ1,2,2), v=[131]
- P(3,0,โ1), v=[2โ13]
- (i) Q(5,โ1,2) (ii) Q(1,1,โ4).
Let u=[3โ10], v=[401], and w=[โ115]. In each case, find x such that:
- 3(2u+x)+w=2xโv
- 2(3vโx)=5w+uโ3x
- x=uโ6v+5w=[โ26419]
Let u=[112], v=[012], and w=[10โ1]. In each case, find numbers a, b, and c such that x=au+bv+cw.
x=[2โ16] x=[130]
- [abc]=[โ586]
Let u=[3โ10], v=[401], and z=[111]. In each case, show that there are no numbers a, b, and c such that:
- au+bv+cz=[121]
- au+bv+cz=[56โ1]
-
If it holds then [3a+4b+cโa+cb+c]=[x1x2x3].
[341x1โ101x2011x3]โ[044x1+3x2โ101x2011x3]If there is to be a solution then x1+3x2=4x3 must hold. This is not satisfied.
Given P1(2,1,โ2) and P2(1,โ2,0). Find the coordinates of the point P:
- 15 the way from P1 to P2
- 14 the way from P2 to P1
- 14[5โ5โ2]
Find the two points trisecting the segment between P(2,3,5) and Q(8,โ6,2).
Let P1(x1,y1,z1) and P2(x2,y2,z2) be two points with vectors p1 and p2, respectively. If r and s are positive integers, show that the point P lying rr+s the way from P1 to P2 has vector
p=(sr+s)p1+(rr+s)p2
In each case, find the point Q:
- \longvectPQ=[20โ3] and P=P(2,โ3,1)
- \longvectPQ=[โ147] and P=P(1,3,โ4)
- Q(0,7,3).
Let u=[20โ4] and v=[21โ2]. In each case find x:
- 2uโโvโv=32(uโ2x)
- 3u+7v=โuโ2(2x+v)
- x=140[โ20โ1314]
Find all vectors u that are parallel to v=[3โ21] and satisfy โuโ=3โvโ.
Let P, Q, and R be the vertices of a parallelogram with adjacent sides PQ and PR. In each case, find the other vertex S.
- P(3,โ1,โ1), Q(1,โ2,0), R(1,โ1,2)
- P(2,0,โ1), Q(โ2,4,1), R(3,โ1,0)
- S(โ1,3,2).
In each case either prove the statement or give an example showing that it is false.
- The zero vector 0 is the only vector of length 0.
- If โvโwโ=0, then v=w.
- If v=โv, then v=0.
- If โvโ=โwโ, then v=w.
- If โvโ=โwโ, then v=ยฑw.
- If v=tw for some scalar t, then v and w have the same direction.
- If v, w, and v+w are nonzero, and v and v+w parallel, then v and w are parallel.
- โโ5vโ=โ5โvโ, for all v.
- If โvโ=โ2vโ, then v=0.
- โv+wโ=โvโ+โwโ, for all v and w.
- T. โvโwโ=0 implies that vโw=0.
- F. โvโ=โโvโ for all v but v=โv only holds if v=0.
- F. If t<0 they have the opposite direction.
- F. โโ5vโ=5โvโ for all v, so it fails if vโ 0.
- F. Take w=โv where vโ 0.
Find the vector and parametric equations of the following lines.
- The line parallel to [2โ10] and passing through P(1,โ1,3).
- The line passing through P(3,โ1,4) and Q(1,0,โ1).
- The line passing through P(3,โ1,4) and Q(3,โ1,5).
- The line parallel to [111] and passing through P(1,1,1).
- The line passing through P(1,0,โ3) and parallel to the line with parametric equations x=โ1+2t, y=2โt, and z=3+3t.
- The line passing through P(2,โ1,1) and parallel to the line with parametric equations x=2โt, y = 1, and z = t.
- The lines through P(1, 0, 1) that meet the line with vector equation \mathbf{p} = \left[ \begin{array}{r} 1\\ 2\\ 0 \end{array} \right] + t \left[ \begin{array}{r} 2\\ -1\\ 2 \end{array} \right] at points at distance 3 from P_{0}(1, 2, 0).
- \left[ \begin{array}{r} 3\\ -1\\ 4 \end{array} \right] + t \left[ \begin{array}{r} 2\\ -1\\ 5 \end{array} \right]; x = 3 + 2t, y = -1 -t, z = 4 + 5t
- \left[ \begin{array}{r} 1\\ 1\\ 1 \end{array} \right] + t \left[ \begin{array}{r} 1\\ 1\\ 1 \end{array} \right]; x = y = z = 1 + t
- \left[ \begin{array}{r} 2\\ -1\\ 1 \end{array} \right] + t \left[ \begin{array}{r} -1\\ 0\\ 1 \end{array} \right]; x = 2 - t, y = -1, z = 1 + t
In each case, verify that the points P and Q lie on the line.
- \begin{array}[t]{ll} x = 3 - 4t & P(-1,3,0), Q(11,0,3) \\ y = 2 + t & \\ z = 1 - t & \end{array}
- \begin{array}[t]{ll} x = 4 - t & P(2,3,-3), Q(-1,3,-9) \\ y = 3 & \\ z = 1 - 2t & \end{array}
- P corresponds to t = 2; Q corresponds to t = 5.
Find the point of intersection (if any) of the following pairs of lines.
- \begin{array}[t]{ll} x = 3 + t & x = 4 + 2s \\ y = 1 - 2t & y = 6 + 3s \\ z = 3 + 3t & z = 1 + s \end{array}
- \begin{array}{ll} x = 1 - t & x = 2s \\ y = 2 + 2t & y = 1 + s \\ z = -1 + 3t & z = 3 \end{array}
- \left[ \begin{array}{c} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 3\\ -1\\ 2 \end{array} \right] + t \left[ \begin{array}{r} 1\\ 1\\ -1 \end{array} \right]
\left[ \begin{array}{c} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ 1\\ -2 \end{array} \right] + s \left[ \begin{array}{r} 2\\ 0\\ 3 \end{array} \right]
- \left[ \begin{array}{c} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 4\\ -1\\ 5 \end{array} \right] + t \left[ \begin{array}{r} 1\\ 0\\ 1 \end{array} \right]
\left[ \begin{array}{c} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 2\\ -7\\ 12 \end{array} \right] + s \left[ \begin{array}{r} 0\\ -2\\ 3 \end{array} \right]
- No intersection
- P(2, -1, 3); t = -2, s = -3
Show that if a line passes through the origin, the vectors of points on the line are all scalar multiples of some fixed nonzero vector.
Show that every line parallel to the z axis has parametric equations x = x_{0}, y = y_{0}, z = t for some fixed numbers x_{0} and y_{0}.
Let \mathbf{d} = \left[ \begin{array}{c} a\\ b\\ c \end{array} \right] be a vector where a, b, and c are all nonzero. Show that the equations of the line through P_{0}(x_{0}, y_{0}, z_{0}) with direction vector \mathbf{d} can be written in the form
\frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z -z_{0}}{c} \nonumber
This is called the symmetric form of the equations.
A parallelogram has sides AB, BC, CD, and DA. Given A(1, -1, 2), C(2, 1, 0), and the midpoint M(1, 0, -3) of AB, find \longvect{BD}.
Find all points C on the line through A(1, -1, 2) and B = (2, 0, 1) such that \| \longvect{AC} \| = 2 \| \longvect{BC} \|.
P(3, 1, 0) or P(\frac{5}{3}, \frac{-1}{3}, \frac{4}{3})
Let A, B, C, D, E, and F be the vertices of a regular hexagon, taken in order. Show that \longvect{AB} + \longvect{AC} + \longvect{AD} + \longvect{AE} + \longvect{AF} = 3\longvect{AD}.
- Let P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, and P_{6} be six points equally spaced on a circle with centre C. Show that
\longvect{CP}_{1} + \longvect{CP}_{2} + \longvect{CP}_{3} + \longvect{CP}_{4} + \longvect{CP}_{5} + \longvect{CP}_{6} = \mathbf{0} \nonumber
- Show that the conclusion in part (a) holds for any even set of points evenly spaced on the circle.
- Show that the conclusion in part (a) holds for three points.
- Do you think it works for any finite set of points evenly spaced around the circle?
- \longvect{CP}_{k} = -\longvect{CP}_{n+k} if 1 \leq k \leq n, where there are 2n points.
Consider a quadrilateral with vertices A, B, C, and D in order (as shown in the diagram).
If the diagonals AC and BD bisect each other, show that the quadrilateral is a parallelogram. (This is the converse of Example [exa:011062].) [Hint: Let E be the intersection of the diagonals. Show that \longvect{AB} = \longvect{DC} by writing \longvect{AB} = \longvect{AE} + \longvect{EB}.]
Consider the parallelogram ABCD (see diagram), and let E be the midpoint of side AD.
Show that BE and AC trisect each other; that is, show that the intersection point is one-third of the way from E to B and from A to C. [Hint: If F is one-third of the way from A to C, show that 2\longvect{EF} = \longvect{FB} and argue as in Example [exa:011062].]
\longvect{DA} = 2\longvect{EA} and 2\longvect{AF} = \longvect{FC}, so 2\longvect{EF} = 2(\longvect{EF} + \longvect{AF}) = \longvect{DA} + \longvect{FC} = \longvect{CB} + \longvect{FC} = \longvect{FC} + \longvect{CB} = \longvect{FB}. Hence \longvect{EF} = \frac{1}{2}\longvect{FB}. So F is the trisection point of both AC and EB.
The line from a vertex of a triangle to the midpoint of the opposite side is called a median of the triangle. If the vertices of a triangle have vectors \mathbf{u}, \mathbf{v}, and \mathbf{w}, show that the point on each median that is \frac{1}{3} the way from the midpoint to the vertex has vector \frac{1}{3}(\mathbf{u} + \mathbf{v} + \mathbf{w}). Conclude that the point C with vector \frac{1}{3}(\mathbf{u} + \mathbf{v} + \mathbf{w}) lies on all three medians. This point C is called the centroid of the triangle.
Given four noncoplanar points in space, the figure with these points as vertices is called a tetrahedron. The line from a vertex through the centroid (see previous exercise) of the triangle formed by the remaining vertices is called a median of the tetrahedron. If \mathbf{u}, \mathbf{v}, \mathbf{w}, and \mathbf{x} are the vectors of the four vertices, show that the point on a median one-fourth the way from the centroid to the vertex has vector \frac{1}{4}(\mathbf{u} + \mathbf{v} + \mathbf{w} + \mathbf{x}). Conclude that the four medians are concurrent.