RECTANGULAR COMPONENTS OF A VECTOR (WITH EXERCISES) - MATHS - 2024

The rectangular components of a vector are the data that make up that vector. To determine them, it is necessary to have a coordinate system, which is generally the Cartesian plane.

Once you have a vector in a coordinate system, you can calculate its components. These are 2, a horizontal component (parallel to the X axis), called "component on the X axis", and a vertical component (parallel to the Y axis), called "component on the Y axis".

Graphical representation of the rectangular components of a vector

In order to determine the components, it is necessary to know certain data of the vector such as its magnitude and the angle that it forms with the X axis.

How to determine the rectangular components of a vector?

To determine these components, certain relationships between right triangles and trigonometric functions must be known.

In the following image you can see this relationship.

Relationships Between Right Triangles and Trigonometric Functions

The sine of an angle is equal to the quotient between the measure of the leg opposite the angle and the measure of the hypotenuse.

On the other hand, the cosine of an angle is equal to the quotient between the measure of the leg adjacent to the angle and the measure of the hypotenuse.

The tangent of an angle is equal to the quotient between the measure of the opposite leg and the measure of the adjacent leg.

In all these relationships it is necessary to establish the corresponding right triangle.

Are there other methods?

Yes. Depending on the data that is provided, the way to calculate the rectangular components of a vector can vary. Another widely used tool is the Pythagorean Theorem.

Exercises

The following exercises put into practice the definition of the rectangular components of a vector and the relationships described above.

First exercise

It is known that a vector A has a magnitude equal to 12 and the angle that it makes with the X axis has a measure of 30 °. Determine the rectangular components of said vector A.

Solution

If the image is appreciated and the formulas described above are used, it can be concluded that the component in the Y axis of vector A is equal to

sin (30 °) = Vy / 12, and therefore Vy = 12 * (1/2) = 6.

On the other hand, we have that the component on the X axis of vector A is equal to

cos (30 °) = Vx / 12, and therefore Vx = 12 * (√3 / 2) = 6√3.

Second exercise

If vector A has a magnitude equal to 5 and the component on the x-axis is equal to 4, determine the value of the component of A on the y-axis.

Solution

Using the Pythagorean Theorem, we have that the magnitude of vector A squared is equal to the sum of the squares of the two rectangular components. That is, M² = (Vx) ² + (Vy) ².

Substituting the given values, you have to

5² = (4) ² + (Vy) ², therefore, 25 = 16 + (Vy) ².

This implies that (Vy) ² = 9 and consequently Vy = 3.

Third exercise

If vector A has a magnitude equal to 4 and it makes an angle of 45 ° with the X axis, determine the rectangular components of that vector.

Solution

Using the relationships between a right triangle and the trigonometric functions, it can be concluded that the component on the Y axis of vector A is equal to

sin (45 °) = Vy / 4, and therefore Vy = 4 * (√2 / 2) = 2√2.

On the other hand, we have that the component on the X axis of vector A is equal to

cos (45 °) = Vx / 4, and therefore Vx = 4 * (√2 / 2) = 2√2.

References

1. Landaverde, FD (1997). Geometry (Reprint ed.). Progress.
2. Leake, D. (2006). Triangles (illustrated ed.). Heinemann-Raintree.
3. Pérez, CD (2006). Precalculation. Pearson Education.
4. Ruiz, Á., & Barrantes, H. (2006). Geometries. Technological of CR.
5. Sullivan, M. (1997). Precalculation. Pearson Education.
6. Sullivan, M. (1997). Trigonometry and Analytical Geometry. Pearson Education.