Learn – Algebra/Precalculus – Vectors

Problem set 1
Problem set 2
Problem set 3
Problem set 4
Problem set 5
Problem set 6
Problem set 7
Problem set 8
Problem set 9
Problem set 10
Problem set 11
Problem set 12
Problem set 13
Problem set 14
Problem set 15
Problem set 16
Problem set 17
Problem set 18
Problem set 19
Problem set 20


Problem set 1

[Work in progress]Problems to test equivalence of directed line segments as vectors, if the line segments are shown in different positions.


Problem set 2

[Work in progress]Problems to add vectors geometrically. Vectors given as directed line segments in different positions.


Problem set 3

[Work in progress]Problems to subtract vectors geometrically. Vectors given as directed line segments in different positions.


Problem set 4

  1. In a river that runs northward, Ben starts swimming with a velocity of \langle -3, 3\rangle.
    1. Will, shortly after, starts swimming at twice the speed of Ben in the same direction as Ben. What is the velocity of Will?
    2. Then, Mishka, starts swimming at three times the speed of Ben in the opposite direction as Ben. What is the velocity of Mishka?
  2. Brady and Derek apply forces of 10 lb and 10\sqrt{3} lb respectively. If the angle between their directions of force application is 90^\circ, determine the magnitude and direction of the effective force on the object.
  3. Grant applies a force of 100 N on an object. James applies a force of 100 N in the direction that makes an angle of 60^\circ with the direction of Grant’s force. Calculate the magnitude and direction of the effective force on the object.
  4. Burkard applies a force of 50 N on an object. Marty applies a certain force on the same object so that the resultant force on the object is 100 N in a direction that makes 60^\circ angle with Burkard’s force. Determine the magnitude and direction of Marty’s force.
  5. Matt applies a force of 100\sqrt{2} N on an object. Freya applies a certain force on the same object so that the resultant force on the object is 100 N in a direction that makes 45^\circ with Matt’s force. Determine the magnitude and direction of Freya’s force.


Problem set 5

  1. A river goes in the north-south direction. Mandy and Ralph cross the river by swimming in the general northwesterly direction, making an angle of 30 degrees with the east-side bank. Mandy swims at 6mph and Ralph swims at 3 mph.They both start and end at the same points.
    1. Show the velocities of Mandy and Ralph graphically on an appropriate reference coordinate plane.
    2. Give the velocity vectors in component form.
  2. Hannah is driving with a velocity of \langle -14.14, -14.14 \rangle. Assume a coordinate plane where the X-axis goes east-west and the Y-axis goes north-south, and assume the velocity’s is given in mph. Describe Hannah’s velocity (magnitude and direction).


Problem set 6

[Work in progress]Problems to give component forms of vectors given as directed line segments in different positions.


Problem set 7

  1. If \mathbf{u} = \langle 1,1 \rangle and \mathbf{v} = \langle -2,2 \rangle, show the following vectors geometrically on the coordinate plane.
    1. 2\mathbf{u}
    2. -\frac{\mathbf{v}}{2}
    3. \mathbf{u} + \mathbf{v}
    4. \mathbf{u} - \mathbf{v}
    5. \mathbf{v} - \mathbf{u}
  2. If \mathbf{u} = \langle -1,\sqrt{3} \rangle and \mathbf{v} = \langle -\sqrt{3},1 \rangle, show the following vectors on the coordinate plane.
    1. \frac{\mathbf{u}}{2}
    2. -\sqrt{3}\mathbf{v}
    3. \mathbf{u} + \mathbf{v}
    4. \mathbf{u} - \mathbf{v}
    5. \mathbf{v} - \mathbf{u}


Problem set 8

  1. Evan is standing 3 feet away from an object and is pulling the object towards him with a force of 20 lb. Carter is standing 3 feet away from the object and 3 feet away from Evan and is pulling the object towards him with a force of 30 lb.
    1. Represent the two forces geometrically. Do not use a coordinate plane.
    2. Geometrically estimate the effective force (magnitude and direction) on the object.
    3. Represent the two forces in component form.
    4. Give the effective force on the object in component form.
  2. Rich, Greg, Brian, Sameer, Arthur and Will are standing in a circle in that order going clockwise. The distances between any two adjacent people are the same. There is an object located at the center of the circle that each one is applying force on to pull it towards himself. They apply forces of 60,50,40,30,20,10 lbs respectively.
    1. Represent the forces geometrically. Do not use a coordinate plane.
    2. Geometrically estimate the effective force (magnitude and direction) on the object.
    3. Represent the forces in component form.
    4. Give the effective force on the object in component form.
  3. Steven is standing 10 feet away from a sturdy desk, and pulls on a rope tied to the desk with a force of 200N. Alex pulls on a second rope tied to the desk also with a force of 200N. The desk can take a maximum force of 200\sqrt{3}N. Determine the distance between Steven and Alex so that they maximize the effective force if
    1. Alex is standing 10 feet away from the object.
    2. Alex is standing 20 feet away from the object.
  4. In a river that flows northward, Bala attempts to swim at a speed of 2mph in the direction that makes 45^\circ with the west bank. At the end of the swim, he finds that his effective speed was actually 2\sqrt{2}mph.
    1. Geometrically, find the speed of the water flow in the river. Do not use coordinate plane or component form for this step.
    2. Geometrically, find the direction in which Bala ended up swimming. Do not use coordinate plane or component form for this step.
    3. Give the component form for the effective velocity of Bala.


Problem set 9

  1. Evaluate the following.
    1. \|\langle 3,-4\rangle\|
    2. \|\langle 3\cos 27^\circ, 3\sin 27^\circ\rangle\|
    3. \|\langle 7\cos \frac{31\pi}{6}, 7\sin \frac{31\pi}{6}\rangle\|
    4. \|\langle 11\cos \frac{\pi}{7}, 11\sin \frac{\pi}{7}\rangle\|
  2. Write 2\mathbf{i}-3\mathbf{j} in component form.
  3. Write \langle -5,-4\rangle in terms of the standard unit vectors.


Problem set 10

For each of the following vectors, find a unit vector in the direction of the vector.

  1. \langle 5,-12\rangle
  2. -3\mathbf{i}-4\mathbf{j}
  3. \langle\sqrt{41}\cos 30^\circ, \sqrt{41}\sin 30^\circ \rangle


Problem set 11

  1. Evaluate \langle 1,-3 \rangle\cdot\langle -2,5\rangle
  2. Evaluate \langle -1,-2 \rangle\cdot\langle -2,-1\rangle


Problem set 12

If \mathbf{u}, \mathbf{v} and \mathbf{w} represent vectors and k is a scalar, justify each of the following.

  1. \mathbf{u}+\mathbf{v} = \mathbf{v}+\mathbf{u}
  2. k(\mathbf{u}+\mathbf{v}) = k\mathbf{u}+k\mathbf{v}
  3. \mathbf{0} + \mathbf{u} = \mathbf{u}
  4. \mathbf{u} - \mathbf{v} = (-1)(\mathbf{v} - \mathbf{u})
  5. \|a\mathbf{u}\| = |a|\,\|\mathbf{u}\|


Problem set 13

If \mathbf{u}, \mathbf{v} and \mathbf{w} represent vectors and k is a scalar, justify each of the following.

  1. \mathbf{u}\cdot\mathbf{v} = \mathbf{v}\cdot\mathbf{u}
  2. \mathbf{u}\cdot(\mathbf{v}+\mathbf{w}) = \mathbf{u}\cdot\mathbf{v}+\mathbf{u}\cdot\mathbf{w}
  3. (\mathbf{u}+\mathbf{v})\cdot\mathbf{w} = \mathbf{u}\cdot\mathbf{w}+\mathbf{v}\cdot\mathbf{w}
  4. (k\mathbf{u}) \cdot\mathbf{v} = k(\mathbf{u}\cdot\mathbf{v})
  5. \mathbf{u}\cdot (k\mathbf{v}) = k(\mathbf{u}\cdot\mathbf{v})
  6. \mathbf{u}\cdot \mathbf{u} = \|\mathbf{u}\|^2


Problem set 14

In the following, \mathbf{u} and \mathbf{v} represent vectors.

  1. If \|\mathbf{u}\| = 5 and \|\mathbf{v}\| = 3, find (\mathbf{u}+\mathbf{v})\cdot(\mathbf{u}-\mathbf{v}).
  2. If \|\mathbf{u}\| = 1 and \|\mathbf{v}\| = 4 and \mathbf{u}\cdot\mathbf{v} = -2, find \|3\mathbf{u} - \mathbf{v}\|.
  3. If \|\mathbf{u}\| = 1 and \|\mathbf{v}\| = 2 and \|\mathbf{v}-3\mathbf{u}\|=1, find \mathbf{u}\cdot\mathbf{v}.
  4. If \|\mathbf{u}\| = 2 and \|\mathbf{v}\| = 1 and \|2\mathbf{u}+3\mathbf{v}\|=6, find \mathbf{u}\cdot\mathbf{v}.


Problem set 15

  1. Find the angle between the two vectors \langle 1,\sqrt{3}\rangle and \langle 9,3\sqrt{3} \rangle.
  2. Find the angle between the two vectors \langle -3,\sqrt{3}\rangle and \langle 9,-9\sqrt{3} \rangle.


Problem set 17

  1. Line l goes through points (6,8\sqrt{3}) and (3,7\sqrt{3}). Line p goes through points (10\sqrt{3}, 7) and (9\sqrt{3}, 8). Determine the angle formed between the lines at their point of intersection.
  2. [Work in progress]Two rectangles adjacent problem with side lengths 1,5 and 2,3.


Problem set 18

    [Work in progress]Problems to find angle between vectors geometrically. Vectors are shown as directed line segments, possibly in different positions.


Problem set 19

    [Work in progress]Problems to find dot product between vectors geometrically. Vectors are shown as directed line segments, possibly in different positions.


Problem set 20

A theorem on vectors says that if \theta is the angle between two vectors \mathbf{u} and \mathbf{v}, \cos \theta = \frac{\mathbf{u}\cdot \mathbf{v}}{\|u\|\|v\|}. Use this theorem to prove the following.

  1. If two vectors \mathbf{u} and \mathbf{v} are orthogonal, then they are perpendicular.
  2. If two vectors \mathbf{u} and \mathbf{v} are perpendicular, then they are orthogonal.

Using the above, can you conclude that perpendicularity and orthogonality are equivalent notions in 2-D space?