Learn – Algebra/Precalculus – Vectors

Representations

Geometric

Exercises

Problem set

In each of the following problems, give which of the line segments represent vectors.






Problem set

In each of the following problems, identify collections of directed line segments that represent the same vector.







Problem set
  1. Reina applies a force of 150 newtons at an angle of 30^\circ East of North. Erica applies a force of 300 newtons at an angle of 60^\circ West of North. Represent each vector geometrically using two different directed line segments.


Algebraic

Exercises

Problem set

In each of the following problems, draw the standard representation of the vector, and write the vector in component form.













Problem set
  1. Write 2\mathbf{i}-3\mathbf{j} in component form.
  2. Write \langle -5,-4\rangle in terms of the standard unit vectors.

Magnitude and direction

Exercises

Problem set

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\|

Problem set

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

Operations

Addition

Exercises

Problem set

In each of the following problems, add the vectors shown and draw the resultant vector.









Problem set

In each of the following problems, add the vectors shown and draw the resultant vector.





Problem set

In each of the following problems, add the vectors shown and draw the resultant vector.





Problem set
  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}

Scalar multiplication

Exercises

Problem set

Subtraction

Exercises

Problem set

In each of the following problems, draw the vectors $\vec{u}-\vec{v}$ and $\vec{v}-\vec{u}$.





Dot product

Exercises

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

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}.

Cross product

Exercises

Problem set

Applications

Angles and projections

Exercises

Problem set
  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
  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.

Word problems

Exercises

Problem set
  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
  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
  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.

Coordinate geometry

Exercises