Welcome to the exciting exercises on vectors and scalar product! Vectors are fundamental tools in the field of mathematics and physics, used to represent directional quantities in the plane and in space. They are especially useful when we want to describe movements, forces, velocities, and many other physical quantities.

In these exercises, you will explore the manipulation of vectors and learn to perform key operations, such as vector addition and subtraction. You will also venture into the exciting world of the scalar product, an operation that combines the magnitudes and directions of two vectors to obtain a numerical value.

Through challenging practical problems, you will test your skills and discover how these mathematical tools are applicable in various disciplines, such as engineering, physics, computer science, computer graphics, and more. So get ready to develop your vector manipulation skills and enhance your ability to solve complex problems.

Solve the following problems:

1

Find the symmetric point of with respect to

Solution

1 We calculate the symmetric point , for which we use

 

 

2 We equate the coordinates and solve for the variables

 

For the first coordinate:

 

 

For the second coordinate:

 

 

3 The symmetric point has coordinates:

2

Given two vertices of a triangle and the centroid , calculate the third vertex.

Solution

1 The formula for the centroid of a triangle with vertices is

 

 

2 We calculate the centroid with the third vertex by substituting in the previous formula:

 

 

3 We equate the coordinates and solve for the variables

 

For the first coordinate:

 

 

For the second coordinate:

 

 

4 The third vertex is:

 

3

Given the points and find a point aligned with and , such that

Solution

1 Since  , we substitute the values of and and obtain:

 

 

2 We equate the coordinates and solve for the variables

 

For the first coordinate:

 

 

For the second coordinate:

 

 

3 The point sought is

4

Calculate the coordinates of so that the quadrilateral with vertices and is a parallelogram.

Solution

1 To find the coordinates , we use the fact that, since the opposite sides of the parallelogram are equal, their vectors are also equal:

 

 

2 We substitute the data and obtain:

 

 

3 We equate the coordinates and solve for the variables

 

For the first coordinate:

 

 

For the second coordinate:

 

 

4 The vertex sought is

5

If form an orthonormal basis, calculate:

a

 

b

 

c

 

d

Solution

1 Since are orthonormal, they are perpendicular to each other, so they form an angle of and their magnitude is 1, that is,

 

2 To find the requested products, we use the formula:

 

 

where is the angle between and

 

3 We find the requested products by substituting into the formula and using the appropriate value of : if the vector is the same and if they are different.

 

a

 

b

 

c

 

d

6

Given the vectors , calculate so that vectors are:

a. Perpendicular.

b. Parallel.

c. Form an angle of .

Solution

a. Perpendicular: Two vectors are perpendicular if their product is zero.

We perform the product and solve for the variable :

 

 

b. Parallel: Two vectors are parallel if their components are proportional, that is:

 

 

We perform the proportion equality and solve for the variable :

 

 

c. Form an angle of : We substitute the values into the vector product formula:

 

 

where

 

 

We square both sides and simplify:

 

 

We solve using the quadratic formula:

 

The roots of the quadratic equation are , but only satisfies the original equation without squaring, so this is the value of sought.

7

Calculate the value of knowing that and

Solution

1 We calculate the vector product:

 

 

2 We equate the result to and solve for :

 

8

Assuming that with respect to the orthonormal basis of the plane, calculate the value so that the vectors and are orthogonal.

Solution

1 Since the basis is orthonormal, we have:

 

 

2 We calculate the dot product of and

 

 

3 Two vectors are orthogonal if their dot product is zero. We substitute and solve for :

 

9

Calculate the projection of the vector onto the vector .

Solution

1 The formula for the projection of vector   onto vector is given by:

 

 

2 We calculate the product of the vectors:

 

 

3 We calculate the magnitude of vector

 

 

4 We substitute the data into the projection formula:

 

10

Find a unit vector in the same direction as the vector

Solution

1 The formula for a unit vector is given by:

 

 

2 We calculate the magnitude of vector :

 

 

3 We substitute into the unit vector formula:

 

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Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.